The short (and admittedly quite boring) answer is that both the Klingon and Romulan Empires are "Empires" because they're run, at least nominally by Emperors.
Although we know that the real power in Romulan society rests with the Praetors and Senate, there are glancing references to a Romulan Empress, for example in Voy : The Q and the Grey
Q: I know that you're probably asking yourself, why would a brilliant, handsome, dashingly omnipotent being like Q want to mate
with a scrawny little bipedal specimen like me?
JANEWAY: Let me guess. No one else in the universe will have you.
Q: Nonsense. I could have chosen a Klingon Targ, the Romulan Empress, a Cyrillian microbe.
The Klingons appear to have gone through a period where there was no extant Emperor. We know from the dialogue in TNG "Rightful Heir" that they reinstated the position with the Kahless Clone:
WORF : You were right about one thing, Koroth... our people are becoming corrupt and decadent... they need moral leadership. Kahless
can be that leader... as Emperor.
GOWRON : (outraged) There hasn't been an Emperor in three centuries!
Notably, there's no discussion of whether such a thing is legally possible, merely whether it's politically desirable to put someone back into the position.