Explanation: we in the modern day differentiate between the Roman Republic and the Roman Empire. The Romans, however, regarded themselves as remaining a republic even with the Emperor at the head. There was a very strong taboo against the Emperor claiming to be king which would not be broken until the fall of the Western Empire. According to the Romans, the Emperor was just a very powerful magistrate who represented the will of the people, and who made decisions in consensus with the Senate.
Provincials sometimes did not appreciate this distinction - Greek cities would often write praises or requests to the Emperor, calling him 'Basileus' ('king'), and receive responses from Imperial secretaries thanking them for their praises or concerns - but insistently in the name of the 'autokrator' ('ruler'), not 'Basileus'.
technically any state is a republic unless it's explicitly a monarchy/polyarchy.
And Roman succession was quite often (mostly?) by inheritance. The "five good emperors" were the big exception, but even they had a habit of adopting their successors.
I think it would be better to see it as a form of nepotism than monarchy in the cases of the Principate, especially considering the ideological considerations of adoption in Roman society. The Senate could (and in the case of Lucius Verus, did, or tried to until Marcus Aurelius threatened to resign) credibly refuse to appoint an Emperor to the position on the grounds that they did not approve of him. They have a very magisterial process of the exercise of Imperial power, even if it is, effectively, autocratic.
I would note that the word Republic comes from Latin ('Res Publica'), and that the Romans regarded it as a unique form of government itself - one which concerned itself with the common good ('Res Publica' - 'a matter of the public') and not just the good of its decision-makers - something which belonged to the people, or at least to general society. Theoretically. Obviously there is a significant gap in execution. Also, that monarchy in other cultures of the period was extremely common.
With that in mind, I think the distinction is important, not just in the Roman conception of themselves, but also in the way we should view the justifications of the state apparatus.
