Not his "personal property," no, but Octavian set up the administration of Egypt to be special. It was possible for him to do so because after Actium and the defeat of Marcus Antonius in 31 BCE, there was no one left standing who could stop him from having his way (more in less) in the capital.

Egypt had proven to have deep, deep pockets, and had proved a dangerous source of revenue for Octavian's rival Antonius. Virtually everything Octavian did in the early years after Actium was designed to eliminate the possibility of another rival emerging and thus another civil war. Egypt also had a lot of grain that had been, up until now, outside of Roman circulation. The principes of Rome had come to realize that controlling and ensuring the grain supply in Rome was very important to maintaining order and control in the capital. Egypt was sort of a "new" source, though that's not entirely accurate.

Given these two points, Octavian moved to secure Egypt for the Romans in a way that had not been done before. To address the first point, he made Egypt an official Roman province in 30 BCE, just months after Actium, BUT: it would not be a normal province, governed by a senatorial proconsular official. Octavian arranged it so that he, personally, would be responsible for the organization and governance of the province. This prevented any dangerous senators from going there as governor and having access to Egypt's economic and agricultural resources. In practice, the Emperors appointed equestrian-rank prefects to rule in Egypt. These were hand-picked men who, in theory, would not entertain thoughts of becoming a rival to the Roman imperial court (but some actually were, like the first prefect Gallus).

Under "Roman" (really Imperial court) rule, Egypt went down a different path from most other provinces. Many of the old Greek ways of doing things remained in place, and a lot of Egyptian-specific practices were preserved and encouraged. But, meanwhile, the infrastructure was massively enhanced to boost economic and agricultural production. The Romans were very good at turning provinces into production machines, and Egypt was no different. A lot of times you will see Egypt referred to as the Emperor's private playground, but that is probably misguided. It was just a powerful place as far as production that was made even better by Roman engineering, and was governed in a special way to ensure that production remained under Imperial control.

Another factor was the fact that Egypt was still hostile to the Romans in 30 BCE, especially up (down) the Nile and on the coast of the Red Sea. This meant that the first decades were spent fighting, and that meant legions. Octavian had already figured out that putting potential senatorial rivals in charge of legions was a bad idea. This is how we wind up with the system of "senatorial" and "imperial" provinces -- the provinces with legions (Syria, Gaul, Spain, for example) would be controlled by imperial prefects, essentially the same system as Egypt. The rest would continue to be governed by senatorial proconsuls, just like during the Republic. What sets Egypt apart from these other "imperial" provinces are the specifics of its administration (many Greek and Egyptian elements) and the specific connection of the place to the Imperial court.