Yes, a PC-lycanthrope can multiattack. Under any interpretation some of them do, and I argue that all do. (And you can skip to the last paragraph if you don't need to see the argument.)

Like you said, The PC-lycanthrope gains all of the abilities that don't involve equipment. A very strict reading would say that any action-paragraph that mentions equipment isn't imported: under this reading the wereboar and wererat import Multiattack just fine, but the other lycanthropes all don't import Multiattack.

This reading is too strict to make sense. It has the effect of making some afflicted with lycanthropy faster and able to multiattack (e.g. wereboar), but some not (e.g. werebear). This, even though all monsters with lycanthropy can multiattack.

Contrast this with a reading in which each portion of an action-paragraph is evaluated against "involves equipment."¹ The effect of this reading is that the PC-lycanthrope gains natural attacks and the ability to multiattack, but doesn't gain any equipment.

A few examples, all quotes from the 5e MM pp.208-11:

A wereboar gains Tusks, doesn't gain Maul, and Multiattack reads as-is:

Multiattack. The wereboar makes two attacks, only one of which can be with its tusks.

A werebear gains Bite, Claw, doesn't gain Greataxe, and Multiattack can be read like:

Multiattack. In bear form, the werebear makes two claw attacks. In humanoid form, it makes two greataxe attacks. In hybrid form it can attack like a bear or a humanoid.

A werewolf gains Bite, Claws, doesn't gain Spear, and Multiattack reads like:

Multiattack (Humanoid or Hybrid form only). The werewolf makes two attacks: one with its bite and one with its claws or spear.

But even this is a little too much lawyering for me: note that the wereboar or werebear both get to make multiattacks that involve weapons, but the werewolf doesn't. Why? Because we're over-thinking it.

The action is the bold-italic word or few words: the paragraph that follows just explains it. (See also AngryGM's article on abbreviating stat-blocks.²) Under this philosophy a PC-wereboar gains Tusks and Multiattack. A PC-werebear gains Bite, Claw, and Multiattack. And a PC-werewolf gains Bite, Claws, and Multiattack. I then understand that wereboar can attack with tusks and a weapon, but not twice with tusks. Werewolves can attack twice--one bite and one that somehow uses the hands--but the PC didn't gain a were-spear when afflicted.

¹ - Note that for all of the lycanthropes in the MM, the only actions that do mention equipment are those relating to their (artificial) weapon attacks.

² - Be warned that AngryGM's posts are liberally sprinkled with profanity and/or rude language. Nothing you might not hear on television, but still worth knowing before you innocently head over there.

Numerous spells, abilities, and actions change

The boss is stronger in strange ways, because they...

• Can try to save four times per round against hold monster and similar save-or-suck spells. Other effects, like a monk's Stunning Strike, are unaffected.
• Can grapple or shove four times per round. (One cannot substitute a grapple for an attack within Multiattack, unlike Extra Attack.)
• Are less affected by one bad initiative roll. A good initiative roll isn't nearly as important as avoiding being the last creature to act.
• Can stand up from prone four times per round.
• Negate surprise after just one of their four turns in the first round.
• Have four chances to beat the Assassin rogue, thus eliminating the rogue's advantage.
• Can Disengage exactly when they need to, while only using one of their four attacks.
• Can cast four spells if they have spellcasting ability. Typically, casting a one action spell takes the place of all of a creatures attacks.

The boss is weaker in strange ways, because they...

• Are forced to save four times per round or take damage from spirit guardians and similar damage spells.
• Are much more quickly affected by save-X-times-or-suck spells, like contagion and flesh to stone.

Legendary Actions are the normal answer

As the other answers stated, Legendary Actions avoid all of these problems by allowing a Legendary creature to act outside of its own turn. An Adult Red Dragon is one such Legendary creature and has the following Legendary Actions:

Legendary Actions

Can take 3 Legendary Actions, choosing from the options below. Only one legendary action can be used at a time, and only at the end of another creature's turn. Spent legendary actions are regained at the start of each turn.

Detect: The dragon makes a Wisdom (Perception) check.

Tail Attack: The dragon makes a tail Attack.

Wing Attack (Costs 2 Actions): The dragon beats its wings. Each creature within 10 ft. of the dragon must succeed on a DC 22 Dexterity saving throw or take 15 (2d6 + 8) bludgeoning damage and be knocked prone. The dragon can then fly up to half its flying speed.

Legendary Actions can be anything. A high-level Legendary spellcaster could fire off four spells per round. A Legendary demon lord could teleport around the battlefield, striking PCs who thought they were safe. A Legendary archdruid could rapidly switch between different beasts, each one making a different attack.

One more thing - if you don't want your boss to fall to the first spell that the control wizard casts, don't forget to add Legendary Resistances. They may feel cheap when you use them, but allowing the boss to auto-save a couple of times helps the encounter be memorable in the right way. Turn 1 banishment against a creature with 8 CHA shouldn't be the answer to everything...