Augustus did everything in his power to ensure that people saw no difference between his rule and the old rule of the senate. It’s unlikely that the average Roman citizen noticed anything was drastically amiss from how the republic used to run.

This is largely due to the fact that the republic was an old and largely false ideal by this point in time—we’ll say 30 BCE for clarity—and very few people if no people who were alive had experienced living in it as the ideal that they conceived it to be. Augustus made sure that his biggest naysayers died in a series of proscriptions a few years prior: these brought him massive amounts of income while eliminating those who spoke against what he was trying to do. This has to be added on top of the fact that the Roman state had known a series of civil wars for generations now, almost 60 years; Sulla marched on Rome in 88 BCE and since then the republic was characterized by a few big name individuals rigging elections, muscling themselves into power, and lording it over everyone else with their wealth and prestige, including the senate. Over time this became normalized for Roman citizens; while it was still an unwanted an uncomfortable situation, it was simply how things were. There’s a fantastic story in Suetonius (which is probably apocryphal, but illustrates the spirit of the times) in which two men each trained a crow (or a raven, I can never remember which) to speak. One trained his to greet Caesar (Octavian) when he arrived in Rome victorious after Actium, and the other was trained to greet Antony. Octavian was lighthearted upon seeing the bird which greeted him and offered the man 20k sesterces for it (a massive sum). The man produced his partner and told him there was another bird, and the bird croaked “hail Antony our victorious commander!” Octavian was not angry, and told the man simply to share the wealth with his partner. What this shows you, even if it isn’t strictly true, is that people were hyper-normalized by this point to the civil wars in which anyone could win, and it made very little difference to most people who did so long as they (thank the gods!) weren’t called up to fight other Romans again.

So, for most Roman citizens, having a big man calling the shots as a consul WAS the republic, since the majority of citizens only had a loose understanding of the republic’s specific laws, and noticed that, for all intents and purposes, this was just how the world around them worked now.

You must also take into account that there was little centralized justice in Rome compared to now. If you wanted protection you couldn’t normally go to a police force, since such an institution didn’t exist. You had to protect yourself and normally this involved latching onto a wealthy patron as their client. You would offer your services to them and they in exchange would protect you with their influence; usually this service involved your vote as well. Thus the Roman world came to be run almost like one massive mafia.

THAT was the republic most people knew, not a representative government designed to keep out kings and keep people free. That was it’s ideal, but few people cared about the ideal. The ancient world was deadly and most people cared about surviving first and foremost, like many people do now. They didn’t care that the republic of 200 BCE was different from now, they cared that someone, finally, seemed to be restoring some order to their land ravaged by the previous civil wars of 6 decades. So long as Augustus made himself look like he was restoring the republic—and this, arguably, was his prime objective in garnering power to himself as a result— most people were content to let him do as he pleased, regardless of whether or not it was strictly legal, because so many “sacred” laws had been broken over and over the last 60 years, and very few people really understood what was going on in the first place. It wasn’t like now where citizens of a republic might be able to veto their leaders and impeach them; almost nobody by the time Augustus establishes himself as the leader of the Roman stage had the ability to power to stop him, and if he kept on going around and saying he’d restored the republic, most people saw the soldiers running around under his orders and agreed with him regardless of whether they wanted to or not.

So, it wouldn’t be until many years later that people would start to recognize that the Roman state was no longer a republic, but an empire with an emperor, and call it as much. (I believe the first “emperor” to be recognized as such by his contemporaries was Vespasian, but someone should fact check me on this, since my knowledge ends mostly after Augustus dies).