More than that, actually. FE1, FE2, and FE3 all had "Hero" classes that actually had essentially the same name even in Japanese as the class that would appear in later FE games as the typical sword/axe class and get translated as Hero. It was written in hiragana rather than kanji, but pronounced the same, so there's no question that it was "supposed" to be the same class. The FE1 and FE3 versions were a typical sword-only promoted class, while the FE2 version was Alm's personal promoted class and used swords and bows.
So yeah what defines the "same" class is very hazy.