I seem to recall seeing, years ago, that Finns have about one third of their DNA ancestry from Asia, and the rest from Europe. That, combined with convergent evolution, would explain a lot of Finnish appearance and some other things.

If the ancient Finns came from Siberia ... well, it's likely that some of the ancient Japanese origins were in Siberia also - or at least directly adjacent to Siberia - so it wouldn't be surprising if there are some odd linguistic similarities there - which might not be coincidental, but nevertheless have had so long to diverge that they seem to have little connection today.