I remember hearing Romans took pride in a fact they don't have kings. And that Caesar would defy kingship. Did they think that emperor is a different entity? What was the important difference for them? Did they acknowledge it's the same thing eventually or were they in denial till the end?
Several Roman writers recognised that there had been a substantial shift with the emperors and related it to the idea of monarchy, including well before Diocletian.
Tacitus, writing about 109 AD, criticised the perversion of republican traditions to the sycophancy of a monarchy.
Suetonius, writing about AD 121, under Hadrian, appears to regard it as a kind of dynastic monarchy, starting from the time of Julius Caesar.
Cassius Dio, writing about AD 200 - 222, openly calls it a monarchy established by Augustus and approves of the change, saying that the Romans could no longer survive under a republic.
(Diocletian came to power in AD 284)
They certainly tried to avoid the appearance of a monarchy/imperial family; Trajan's monument in Dacia speaks of how Roman soldiers fought and fell in defence of the Republic, and he is considered the perfect prince because he supposedly didn't wield strict authoritarian power like Domitian had (although historians like Bennett dispute this idea and argue he was actually just as, if not more, authoritarian)
Although AJc pointed out how Roman society eventually accepted this state of affairs I believe there was some acknowledgement of reality long before then. Lucan, I believe, complained about how, after Caesar defeated Pompey at Pharsalus and thus gained primacy in Rome, liberty could only be found beyond the Rhine and the Euphrates (ie beyond Rome's borders). The Romans certainly accepted that they were under the control of individual men as opposed to a republic, but the facade of republicanism was upheld for quite some time.