Can Beast make multiple attacks if he’s carried multiple times? My gut says yes, but I’d like a second opinion.

It seems like it should work.

The way I am reading the power, he is just being able to be given an action after being carried (attacking in this case), but still takes an action token. Thus, it would be one and done.

But the effect is an instruction not an action.
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My gut reaction was no because he's friendly to himself, but it DOES say "Other" so I think that you'd be good to go on this strategy

As long as no other friendly character has been placed, every time Beast is carried it will trigger his effect.

So if Beast is carried multiple times and no other characters have been placed, it will activate each time.

That is not at all what the effect says. It is a trigger that allows him to make an attack. There is no action of any type involved.

And yes, it can trigger multiple times as long as no other friendly characters have been placed.

Does Beast receive a token for this attack or does it count toward your action total for the turn?

They're attacks, not actions. Since you do not give him action he neither receives a token nor does count towards your action total (since as mention they're not actions).