Well Daniel did name the town after his son and sole heir Lev. And also Daniel and his father and grandfather before him were pretty much as removed from their Scandinavian viking forefathers as Tsars of Moscovia were from their Kievan Rus forefathers. Sure Daniel and his druzhina followers had noble viking/varangian roots but their adopted language was very much old east slavic which basically had elements of both russian and ukrainian. But going back to the point, city name Lviv stems from the sons name Lev which isn't a Scandinavian name at all but actually in modern Ukrainian still means "lion". So Lviv, just is a way of saying "Lev's place" (or literally something that belongs to Lev). But yeah I must agree that in 1200s Kieven Rus probably didnt speak pure Ukrainian dialect (as much as I sometimes like to wish they did
) but the Ruthenian language (forerunner of modern Ukrainian) was already emerging by 1300s right around the time Lviv became Polish.