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(Remotely related to Is there an in-universe explanation for the Roman Empire terminology used for the Romulans?)

The Romulan Empire and the Klingon Empire are led by a senate and a council with a chancellor, respectively. The position as chancellor is not inherited.

How, apart from maybe "Gene Roddenberry had no clue, but wanted something that sounds cool" do these qualify as "empire" at all? An empire should have, well, an emperor (or at least a queen, like the British Empire).

It is a common misunderstanding that empire always implies monarchy or royalty.

The term "emperor" derives from the Latin term imperator. It roughly means "commander" or "protector". This was roughly Julius Caesar's function when he came to power in the Roman Republic circa 50 BC, under the official title dictator. It is true that his successor Octavius (later Augustus) was a blood relative of Caesar, but he did not come to power through bloodline alone. (The Battle of Actium was the deciding factor.) Augustus held the title of imperator, thereby beginning the line of Roman emperors. These emperors did not necessarily share a common dynastic origin.

In short, inheritance is not a necessary part of the ruling of an empire.

From Wikipedia:

An empire is a geographically extensive group of diverse states and peoples (ethnic groups) united and ruled by a central authority, either by a monarch (emperor, empress) or an oligarchy. The term empire is derived from the Latin term imperium (a rule, a command; authority, control, power; supreme power, sole dominion; military authority; a dominion, realm), the 'ruling’ of territories that are far beyond the homeland.

It is clear that both the Klingon Empire and the Romulan Star Empire are consistent with this usage of "empire". The leadership of both entities is oligarchical, with power concentrated in the hands of a few, and with vast Alpha and Beta quadrant territories under their control.

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.

Romulan Empire.

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.

Klingon Empire.

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.

Qo'noS and Romulus have Supreme leaders (not called as such but they exist). Gowron was the supreme authority for a while for the Klingon empire and Shinzo for a while was the head of the Romulan empire. They each had predecessors and successors of course so in effect they do fall under the Empire government structure. S

Well for on an "Emperor" is a retired king as in possibly non sexual status now so just good at knowing what he used to do or life expectancy oriented now normally. See Asian Empires ans such especially obvious they know how its done. Its a certain personality always that is why Europe was so plagued and held back by false crowns like Hitler. Real kings had special spiritual abilities and understandings before they became an emperor for instance. An "empire" is places that used to be "kingdoms" for instance, religiously speaking under the king of kings he was called even still. Palace level is higher than castles. Most were non official for reasons unknown depending. So in galaxies this is the case also. Corruption of em did happen, crude was a result. See the empire state of California and Empire of Japan. Benjamin Sisco of DS9s was a real king even at one time for instance too. Most if not all civilizations had to have some sort of kings involved to survive. More pagan cultures rip each other off and tend to go extinct or move. Perhaps ruled by others eventually cause of it and needed emergency help by others in know how to do it correctly that works better. A higher rank king was also involved in star trek dynamics levels even, probably HMH of the empire state, we think.