There is no reason to view the Mede empire as an continuation of the true Dragonblood Septim Empire, except maybe being obtuse.
The Mede's are a family of Colovian warlords who threathened the Capital with destruction in order to force Council to declare Titus I emperor.
Their line is illegitimate, and has no legal nor moral authority.
Thats true it doesnt act the same but again, that has happened to real life empires in the past, Trajan was a general with no connection to the Royal line of Latin rulers and he took control of Rome but it was still called "The ROman Empire" not "The Trajan Roman EMpire or The Trajan Empire". it could only be seen as another different empire if The Septim Empire completely collapsed, like if idk (hypothetical), Morrowind invaded the capital and saked it then saked Skyrim and High Rock said screw it we quit, if the Medes rebuilt Cyrodil and retook High Rock and Skyrim, THEN it would be a different Empire completely