It can be noted that Finnish and Hungarian are more distant than English and Persian.

What is absolutely astonishing, however, is how similar they sound, at least in terms of prosody. To my Hungarian ears Finnish, at least if heard from distance, sounds eerily familiar.

Compare this to e.g. Spanish and French, languages so close to each other but sounding so different. I wonder if there is some deep reason why prosody is well conserved in the Finnish-Hungarian pair and apparently entirely meaningless in Romance.

In terms of words or grammatical structures?

In terms of words https://histdoc.net/sounds/hungary.html

Note that Altaic itself is considered discredited among linguists, to say nothing of the Uralic-Altaic (which never found serious purchase).

If you look from Indo-European perspective, in other words if you look only at the common words, yes, they share very little common words. And then no one explains all those same grammatical structures shared between those languages. In the article replace Finnish with Turkish, Mongolian, Hungarian or Japanese and again that article will be correct again.

Is there a genetic link between Koreans and Finns?

Well Finland and North Korea are only separated by one country.

There doesn't have to be. A people's language family and genetics don't necessarily have to match, because a small group of conquerors can force their language upon a much larger gene pool. Modern day Hungarians, for example, are genetically only marginally different from their Slavic neighbours.

I have found this one https://en.wikipedia.org/wiki/Haplogroup_N-M231

Hm, I thought Europeans were R1A1

R1a1a seems to correspond to Proto-Indo-European-speaking migrants, but that's not the only group that contributed to the genetics of the present population of Europe.