I'll dissect this homebrew race on a per-trait basis, and then offer an overall evaluation at the end:

Neutral traits

These are the traits I find are neither over or underpowered:

Ability Score Increases. Your Strength score increases by 2, and your Dexterity score increases by 1.

Pretty standard, nothing under or overpowered here.

Age. Lycanthropes mature at the same rate as Humans, and live up to around 150 years. Alignment. A Lycanthrope is typically of a Chaotic Alignment.
Size. Lycanthropes stand 6 to 7 feet tall just as a Human. Your size is Medium.

All fluff.

Speed. Your base walking speed is 30 feet in your humanoid and hybrid forms, and 50 feet in wolf form.

This is better than what most races get, but it's situational enough to not be anything of note.

Darkvision. You can see in dim light within 60 feet as if it were bright light, and in darkness as if it were dim light. You can't discern color in darkness, only shades of gray and red.

Pretty standard. I like the addition of the red color.

Tracking. You can detect opponents within 15 feet by sense of smell. If the opponent is upwind, the range increases to 40 feet; if downwind, it drops to 10 feet. Strong scents can be detected at twice the ranges noted above. Overpowering scents can be detected at triple normal range. When a creature detects a scent, the exact location of the source is not revealed—only its presence somewhere within range.

This is a cool trait, but I'm doubtful as to its actual usefulness in-game. The range is so short that you're still going to get surprised, and tracking whether you're upwind or if the creature is smelly sounds like it will quickly get tedious. Consider giving the Werewolf Keen Smell or Keen Senses (Proficiency in Perception), like an Elf.

Shapeshifter. You can take 1 action to change into your Hybrid or Wolf Form. While in Hybrid Form you can not wield weapons, or wear armor. While in wolf form you cannot wear armour or wield weapons. You are able to wear certain equipment at the DM's discretion, such as circlets and robes. You can choose to have equipment you are wearing to either merge into your form or be dropped to the ground. You gain no benefit from equipment that is merged into your form.

The ability to change, itself, is pretty standard, however, the restriction on weapons and armor are weird (They can wield Foci and Holy Symbols, but not weapons?) This trait is probably the biggest counter-balance to all the positive traits because of the limitation on what you can hold in your hand.

Languages. You know Common, Wolf and one extra language. You can only speak in Human and Hybrid forms.

Pretty Standard, though I wonder why you can't speak Wolf in Wolf form.

Wolf Empathy. You can communicate with wolves and dire wolves.

Exceptional but not overpowered. Gnomes can speak with small animals, which I've seen can be useful at times, this trait is going to be a lot more useful than its gnomish counterpart.

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Positive Traits

These are the traits that your Werewolf race gains as benefits, which I rate as over or underpowered based on comparison with existing traits from other published races:

Regeneration. You have resistance to damage from nonmagical weapons that aren't silvered. At the start of each of your turns, you regain hit points equal to 1d4 + your Constitution modifier if you have no more than half your hit points left, and have at least 1 hit point.

Overpowered. This trait effectively gives the Werewolf race 1.5x to double HP, with the resistance alone. Similar races with resistance such as the Dragonborn (1 element), Dwarf (poison) and the Aasimar (radiant, necrotic), grant only 1 or 2 types of resistances and only for situational damage types- a resistance to the three most common damage types is too strong.

The actual regeneration trait is also very strong. The only race I found with regeneration is the UA-Gothic Heroes Revenant subrace, and the Revenant only regenerates 1 HP per turn, your homebrew has the potential to regenerate 5 HP, with a +3 Con, at 1st-level.

Natural Weapons. While in your hybrid or wolf forms, you have a 1D6 Bite and 1D8 Claw attack. Increase damage at level 5 to Bite 1D8 and Claw 1D10. On a successful melee attack you may use your bonus action to make a claw attack on the same target.

This is tricksy. The d6 Bite is ok, but the claw attack basically grants you access to a non-light one-hand Martial weapon you can use to attack with a Bonus Action, normally you'd need Dual Wielder to pull that off. Taken in conjunction with the Shapeshifter trait where you can't wield anything in your hands when you transform, it sounds ok, but then there's gonna be the powergamer who will want to play a Monk in order to get the early d10 unarmed strikes, with flurry of blows.

Then again, if this race will be paired with any class that isn't a monk, he's gonna have no access to magical weaponry, and be forever at the mercy of creatures with resistance to Bludgeoning, Piercing and Slashing damage.

Trip Attack. While in Wolf or Hybrid form you can attempt to trip the opponent as a bonus action after a bite attack. The opponent must make a Strength saving throw equal to 10+ your Strength modifier + your proficiency bonus.

Flavorful and only slightly overpowered- most save DCs are computed as 8 + modifier + proficiency.

Skills. You are Proficient in Survival, Nature, Intimidation and Perception rolls. You have disadvantage on Persuasion checks.

Overpowered. A Half Elf with Skill Versatility only gets 2 skills, this trait grants four. The Disadvantage with Persuasion doesn't counter balance this enough (nor does it make much sense, do you still get Disadvantage in Human Form?).

Saving Throws. You are proficient with Strength and Dexterity saving throws.

Overpowered. No race grants Proficiency in saving throws. These are strictly benefits from picking a class.

Tough Hide. While you are not wearing any Armour or shields your AC becomes 10+ Strength Modifier + Proficiency Bonus (All Forms)

Overpowered. Unarmored Defenses usually don't include Proficiency Bonus as a component. This is because Proficiency is "free" as the character levels up, and it will quickly get out of hand at higher levels. Consider that at 17th-level, and at +5 strength, this character will have 22 AC, even before items that boost strength like a Belt of Giant Strength.

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Negative Traits

These traits would be the counter-balancing factors for the Werewolf race. I will say, in advance, that these traits are not effective counter-balancers (it's not as discouraging as, say, Sunlight Sensitivity) for all the Positive Traits above. I'll explain why below each one.

Silver Weakness. You are vulnerable to damage from silvered weapons.

Vulnerability is normally really punishing, but in this very specific case, when the vulnerability comes into play, it will feel like you're specifically targeting the PC, As Erik puts so well in another answer.

Also consider that not even actual Lycanthropes in the MM are vulnerable to silvered weapons, and no official race gives you vulnerability to anything.

Full Moon Rage. At the start of a night with a Full Moon you must make a DC13 Wisdom save or lose control of yourself. During this time the DM has Control over what your character does, however every hour you may reroll your saving throw to take back control. Once you make the correct saving throw you retain control for the rest of the night.

Not a huge counter-balance, it will only come up once a month, and the DC is set pretty low (with a chance to break it every hour). What's more, smart players will make preparations if they know the full moon is coming up. They're going to tie themselves to trees or manacle themselves to a cell, which negates what this is supposed to accomplish.

New Moon. During a new moon, you cannot transform, and are damaged normally (without resistances and weaknesses).

As above, it's not a huge counter-balance. Smart players will avoid adventuring at this time of weakness. If you force them to adventure in this scenario, it will feel antagonistic.

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Overall, this race is overpowered, almost broken. What's really glaringly overpowered are the Regeneration, Saving Throws, Tough Hide, Skills, and how the Natural Weapons scale so fast. The Werewolf race will outshine just about anything at low-to-mid-levels, and be powerless against higher-level opponents with resistances.

The race has too many things going for it, if you take a look at other races, they usually have 3 or 4 defining traits (not including ability score bonuses, darkvision, languages, etc), this race has a whopping 8.

Using this race power-level analyzer, from reddit's /u/aranim and /u/JamesMusicus, this Werewolf scores:

• Ability Score Increase +3
• Silver Weakness -1
• Regeneration +2 (from regeneration) +1.5 (from resistances) = +3.5
• Natural Weapons +1
• Tracking +0.5
• Darkvision +0.5
• Shapeshifter +0
• Trip Attack +0.5
• Wolf Empathy +0.5
• Skills +2
• Saving Throws +2
• Full Moon/New Moon weakness -2
• Tough Hide +2 (being conservative)
• Language +0.5

Total 13 points, nearly triple the score of the poor Dragonborn (4.5), and eclipsing the highest-rated Mountain Dwarf (8).

Overall, Looks Good!

Almost all of these abilities have a precedent. Damage resistance is shared by dragonborn. Proficiency in what is essentially a two handed weapon does not currently exist RAW -Dwarves and Elves are only proficient in 1 handed weapons- but I don't think this is a big deal. Any class that isn't proficient in a two handed weapon probably doesn't want to use one anyway.

Rebalancing of Bear Paws

Bear Paws. You have proficiency with your claws and teeth. Your claws deal 2d6+Str slashing damage and your teeth deal 1d8+Str piercing damage. Due to your clumsy paws, you can’t use any artisan's tools, musical instruments, light weapons or finnese weapons.

I agree with Dan B that it would be helpful to parse these out for the sake of readability. I also agree that it's important to prevent the claws from being used as monk weapons. That said, you don't want to remove all flavor: if you bump the bite damage down to 1d6+Str, it will be the same as the Lizardfolk's Bite ability, which is a natural weapon that can be used for monk abilities.

It might also be good to note that these are both "natural weapons" in order to better ground them in the system.

I don't think there's any need to state how many attacks you can make with these weapons per round. None of the natural weapon abilities of RAW classes do this. No weapon description does this. Your claws are weapons, natural weapons - the rules already provide ample guidance on how weapons can be used.

Finally, it is vitally important that you specify that a claw attack requires use of both paws - otherwise, your class could use a shield in one hand and a 2d6 weapon in the other.

Smaller Concerns

The swim speed is perhaps too high, not just from a narrative perspective (a bear that can swim as fast as it can run?) but also because it is the same as the Triton swim speed, and it's probably preferable to not steal the aquatic stage from the fish people race.

From a narrative perspective, I'm curious how the bear can use any weapons at all, but that might be too strong a debuff.

In line with other spellcaster-created creatures from the MM, you might change language from "Common" to "One language known by its creator". Also, maybe "You can speak, read, and with great difficulty write in one language of your creator." would be more appropriate given the trouble using tools?