How come "John is friends with Jane"?

The usage in the question title seems common enough to me, though it may be more common in Britain.

But I can't exactly see what "part of speech" the word friends is here, and I can't come up with any related forms. You'd never hear, for example, John is enemies with Jack. What is going on?

We don’t have ‘make enemies with’, but we do have ‘make enemies of’, just as we have ‘make friends of’, so in ‘friends with’, ‘friends’ would appear to be a noun. (The OED has ‘friends with’ under its definitions of ‘friend’ as a noun.) ‘Friends’, because friendship requires two participants as a minumum. It’s not just friends, of course. We can be ‘pals with’, ‘buddies with’, ‘mates with’, ‘chums with’, ‘partners with’ . . .

Good question. What kind of word class is "friends" in to be/to make/to become friends/good friends with someone"?

The German translation of John is friends with Jane would be John ist mit Jane befreudet. That would be "befriended with" (That word exists in English). English doesn't use "befriended" but it can express the same idea with friends.

One could try to label this "friends" in the mentioned idioms as special adjective limited in use to those idioms. But that jams when "to be good friends with sb" is used. So we have to see "friends" as noun in plural form with special limited use.

I would guess the basis of the idioms is "John and Jane are friends/good friends" and this expression was artfully twisted and transformed into "John is friends with Jane", maybe to replace a "befriended with", which came out of use.

Idioms don 't care whether they are grammatical or not, they have their own rules.

Etymonline doesn't mention the problem in its entry friend. Perhaps someone can have a look at OED, I have no access.