They’re basically mutts...like the Russians. The British are pretty much mutts too.
Mongolian is an Altaic language. Finnish, Estonian, Sami, and Hungarian are Uralic languages. Some decades ago there was a conjectural Ural-Altaic language family, but more recently this theory has been pretty well abandoned.
At the same time the very latest research detaches all 11 modern Sa'ami languages from the Fenno-Ughric sub-group for several reasons. One is that it can now be demonstrated that the so-called "German Loan Words" in Sa'ami actually originated in Sa'ami and were passed to the Germans at an early time.
Also, with 11 clearly identifiable Sa'ami languages (some of which are mutually incomprehensible to the others), it's pretty obvious the Sa'ami languages have been developing in isolation far before contact was made with the East.
The common vocabularies occur only for distinct bodies of technology or agriculture. The Sa'ami, of course, had no agriculture!
BTW, your basic Korean and Tibetan "red heads" must necessarily have a FAR WESTERN EUROPEAN origin ~ so that gives you a good idea where the Mongols got their languages.
Note, it was very common in ancient times for tribes to trade surplus girls for horses, pigs, cows, and other useful items. The result was the passage of both genes and words to other un-related tribes. Given enough time, those genes, and words, could make the trip all the way across Eurasia and back!
That’s the old theory where it was assumed the Western Finno-Ughric languages were assumed to have come in from the East, rather than originating in the West and moving East in the more distant past.