Technically Japanese isn't an isolate: it is part of the Japonic family along with the Ryukyuan languages.

EDIT: Korean is also related to the Jeju language, although the latter is severely endangered.

Maybe because of the mountains?

Honestly I am not convinced the two aren't somehow related. Learning Korean it felt as though a quarter was basically Japanese words with weird pronunciation shifts and another quarter was Chinese in origin. I do get that both languages would have loanwords but for me the overlap felt a bit too intensive to both languages. Almost like a weird puzzle piece linking them.

There are theories that Japanese and Korean might be related but you'll get plenty of deniers on both sides (understandable considering their history) and it seems to be on that cusp where language studies either sides with loan-words and cross-pollination or with interrelatedness. That being said I do think that the Altaic language theory doesn't hold up. Just because there is an overlap from Japanese to Korean and than a questionable one to the Mongolic languages and so on until you land in the Turkic languages is not proof that they are all related. The same logic could be applied to Jp-Kor-Chn and then into Burmese. But Korean as a link barely works in this case and beyond the odd Chinese loanword you'll be hard pressed to find any relationship between Burmese and Japanese as opposite ends of the chain if you will.