Coincidentally, this weekend I’ve been watching a dramatized documentary on Netflix about the Roman Empire. Throughout the Roman Republic era, and the Empire Era later, Rome was very dependent on Egyptian commerce particularly grain. Egypt paid tribute to Rome for centuries. They were allies and then later, about a century after the Anthony and Cleopatra era (apx 100 AD) Egypt became part of the Roman empire.
The de facto rule of Egypt by Rome started with Julius Caesar, who put the unpopular Cleopatra VII on the throne and kept her there until his death; Antony continued the same practice, and since she was right there, he also had some kids with her (unlike Caesarion, they survived and were married off into Roman nobility); Roman occupation of Egypt was one of the easier things they did. But you're right, the form of rule Rome imposed varied throughout the Empire, as a way of transitioning the locals into the idea. Roman/Byzantine rule of Egypt continued until the 7th century conquest by the Mohammedans.
“They were allies and then later, about a century after the Anthony and Cleopatra era”
If Cleo had her way (and she might very well have succeeded),
it would have been an empire centered on Alexandria, and not Rome. Alexandria was a key trade hub between east Asia, India, etc, and Rome. It was much wealthier and occupied a better, safer, geographic position.