My favorite little coincidence is that ending a sentence in "ne" turns it into a question in both Japanese and colloquial German, with exactly the same sense ("isn't it?").
People have mentioned other languages (Polish, Latin, French, English) but the negation particles in all of those languages is from the same source (Proto-Indo-European "ne"). The Japanese one is entirely unrelated though.
Also Portuguese, where "né?" is understood as a contraction of "não é?" ("isn't it?").
I don't mean to say that this is a coincidence as far as the European languages go; negation words with N are often a shared inheritance from Indo-European
Americans (and I'd imagine the British) have "eh?" too, it's just not as common ("not bad, eh?"). Among Americans I feel like I've seen it more in casual written online conversation than in speech.
I've noticed a similar similarity between the particle "yo" in Japanese and adverb "ju" in Swedish, both used for the same purpose at the end of sentences.
A few more: Swedish:"tabberas"(eat everything) from Latin:"tabula rasa"(clean slate) <=> Japanese: "taberu"(eat).
Swedish:"må" <=> Japanese:"mo". "allowed to" Pronounced the same.
Expression: Swedish:"det går inte"(not possible, not allowed to), literally "that does not go" <=> Japanese:"ikemasen"(not allowed to), literally "can not go".