To the question in the title: Eeeh kinda?
There's no such thing as Shouts in the earlier games so the emperors don't go around Fus-Do-Rahing enemies off a cliff (too bad, might have saved the emperor in oblivion from getting killed).
However, the Amulet of Kings can, in lore, only be worn by someone with the blood of a dragonborn, so dragonborns existed and were mentioned. Though I recall in Oblivion the reference was more to that lineage being blessed by Akatosh.
To the question in the the... question, they are all the same universe, set some time apart. Books have survived from the very first game and the concept of time (the dragon) breaking is a thing in all of them (this also explains how all the games are canon despite having such freedom of choice).