This is not balanced with the other races in the PHB.

Assuming that these are two different sub races ...

1. You are starting with a feat.
2. You are starting at size large: no PHB race is large. (Not even Goliath)
   Why this may matter:
   1. Weapon damage changes. You get 1d4 bonus when you go from medium to large (see potion of growth, enlargement spell). There is a reason all player playable races are medium sized. The PHB doesn't explain how damage goes up for a Large sized PC because No PC is sized Large. It's a default.
   2. Monsters get additional damage dice if large. (This should not apply to you, though).
   3. Being large also allows you to Grapple and Shove Huge sized opponents. It also means that small sized monsters can't grapple or shove you (Kobolds, Goblins). That becomes a non-trivial benefit as you fight things larger than large at higher levels. You can ask my half-orc champion versus giants, cyclops, etc, on that score. Without a potion/spell, his shield master shove does not function.
3. Unarmored AC is overly generous, given that class isn't yet given.

4. Natural weapons at 1d6 + str ... significantly more than PHB races. (Though similar to Tabaxi from Volo's ... 1d4)

   The base class benefits of climb, cold resistance, etc fit well enough on their own for the base race. You added a further two skills in the searching for perfection. (Equivalent to half elf skill boost).

Your Proposed bonuses:
For Eir: Dex +2. With two feats. Similar to vHuman.
For Hiojuth: Str and Con + 1. Similar to vHuman.

Compare to variant Human: +1 to two abilities and one feat, and one added skill.

Skills: Your "nimbles claws" as thieves tools is an equivalent to a vhuman added skill, but it isn't a skill. Using it as a bonus action begins to approach a feat. Add to that unarmored defense and you are out of balance.

Add Athletics Proficiency, which makes for three additional skills for the Hiojuth sub-race, isn't too far out of whack. You then added the half-orc racial unique ability for Hiojuth on top of that. And then unarmored defense ...

Natural Armor: natural armor (6 when un armored, ac = 10 + nat armor + dex mod)(maybe changing that to con mod?)

Natural armor as presented is equivalent to roughly plate armor with no shield: 10 + 6 + either Dex or Con mod. (Depends in how high your dexterity is at roll up, to which you have already added 2). If your starting Dex is 16, you have AC 19 without armor. A monk with Wis 16 and Dex 16 has a 16 AC. If you picked a monk, and you have a decent wisdom (14) and 16 dex (Eir), your unarmored AC is 20.

No, this is not balanced.

What to fix?

Size: Medium. Like all other races in PHB and Volo's. (Except for the small ones ...)
Base race: Good! But change Natural Weapons to 1d4 + bonus
Sub races: Lose the feats. Keep the rest.
Eir: Make nimble claws "equivalent to thieves tools" but drop the bonus action.

Unarmored AC: lose it, or trade one of the other base class attributes for a +3 bonus.

Notes.

The unarmored attribute is usually a class ability (Draconic Bloodline sorcerer, Barbarian, Monk, although a few monster races like Tortles or Lizardfolk have something like this).

An argument for "lose it" comes from @Ethan, who made a point in comments about how the game treats Armor Class, and bounded accuracy: the unarmored defense pushes up against a 5E design principle. There are almost no plain AC bonuses. Typically, the game presents an alternate calculation like similar to mage armor. "While not wearing armor, your AC is 12(or 13) + dex mod" would fit the design model more closely. (See the Sorcerer example in the PHB).

The AC changes, to 13

In their human form, the PC weretiger with 16 DEX wears the studded leather armor, giving them 12 + 3 (= 15) AC.

They polymorph into their tiger form.

Any equipment it is wearing or carrying isn't transformed.

As you note, the armor does not transform to fit them, so they do not benefit from the human-shaped armor. It is the DM's discretion whether the armor is destroyed, damaged, or just hangs off the weretiger's new form uselessly after the transformation.

Since they are not benefiting from the armor any more, they can no longer use the AC calculation (12 + DEXMOD) provided by the armor. As Solias notes, in the Player Characters as Lycanthropes box on MM page 207, AC modifications in different forms are noted for other types of lycanthropes, but not for weretigers, so their AC is now simply the basic 10 + DEXMOD = 13.

Equipment is not a statistic

Your confusion seems to be centered around the following passage:

Its statistics, other than its size, are the same in each form.

And this is true.

If the weretiger were wearing a Ring of Fire Resistance which it took off and dropped exactly as it transformed, it would not still have resistance to fire in its tiger form, despite "resistances" being a statistic. This is because fire resistance is not a statistic of the PC, but something provided by the ring to the PC.

The same is true for the AC calculation (12 + DEXMOD) provided by the studded leather armor - it is not a statistic of the PC, but something provided by the armor to the PC.

Jeremy Crawford has tweeted to clarify that equipment is not an inherent part of monsters in the context of a similar question around the Shapechange spell - the same is true of PCs.