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2 being the only even prime isn't really anything fundamentally weird. Every prime is the only divisible-by-that-number prime. 2 has nothing unique about that.

We only notice the case for 2 because our human languages happen to define divisible-by-2 as a word and concept. If our languages called divisible-by-3 "treven" or something like that, we'd think it weird that 3 was the only treven prime.

It’s a little weird. Numbers being their own additive inverse in characteristic-2 makes for some special cases. (But I guess if we did algebra with ternary operators, 3 might be weird too.)

Only if we can declare 3 composite because it's annoying to have a number divisible by 3 in the primes, and so on for the rest of them.

It’s composite in the Gaussian integers, maybe that helps.