Can this sentence mean "We don't know each other." (referring to "I and he")?

Yes.

And it can also mean "We don't know him.", yes? So it's ambiguous?

Yes, it can, but I wouldn't call this ambiguity. It means "We (inclusive, including him) don't know each other". "We" can mean any number of people, from 2 onwards.

This way, all plural sentences can be tagged ambiguous, since we don't know the exact number.

I meant, I thought "мы с ним" is a typical Russian (Slavic?) expression for "I and he" (or a dual, exclusive first person). So my first interpretation of the sentence, when I read it, was "We [I and he] are not familiar with each other.", but reading its translations I thought I might have been wrong, because then I interpreted "с ним" as being an argument of the verb, as in "We are not familiar *with him*."

It can mean either.

«Мы с…» is just a way to express inclusive «we» (though I'm not sure this term applies here).

«Мы с ним» is not neccessarily dual (though it often is), it can mean either «I and he» or «we and he».