"ő" exists in Hungarian, it is pronounced as a longer "ö" (and "ö" is pronounced like the first vowel in the English word "dirty"). But in German, only "ö" exists (and it depends on the actual dialect/context, how long is it pronounced).
"König" is originally a German word, and it means "King". His name has German origin, already the "Kőnig" version was a Hungarianized form (it sounds simply more natural for native speakers).
While in the region and the era was/is it common to use some simplification/translation of the names, mostly to adapt better foreign-speaker environments, he and his father lived their entire life in Hungary and they did not need to do that.
The probable reason today to use the "ö"-Form is that "ö" exists in the most common extension of the ascii encoding (so-named "latin1"), but "ő" does not. Another probable reason was that the typewriters and pre-unicode computers of the era could write "ö" far more easily.
In a German text I would use the "ö"-version on decency reason.
Both the English and the German Wikipedia use the Kőnig name.