The Persian empire comquers Egypt, then Alexander conquers the Persians, and in his titles there's one with the name “pharaoh of Egypt” but did he ever was like one of the other pharaohs? Or how was his story with the egyptian crown?
At the heart of Egyptian politics was religion and vice versa, and as a result, no man could ever rule a happy and willing Egypt without accepting this.
The pharaoh was more than just a king, but an incarnation of Horus, the son of Osiris, god of the afterlife. In order for any pharaoh to be denominated as so, he must be chosen by Amun (the king of the Gods) to rule Egypt. As a result, foreign rulers in Egypt would have priests seek council with the Gods in order to legitimize their rule.
The Hyksos kings in the Second Intermediate Period (Dyn13) were appointed by Amun, as were the Nubian Kings of the 3rd Intermediate period, Cambyses II of Persia sent his general Darius I to keep watch over Egypt in Dyn 27 and he too, sought the approval of the Gods. then Xerxes, Ptolemy etc.
Alexander of Macedonia conquered Egypt from the Persians when he was 24 years old, and Egyptians saw him as a "liberator" who freed them from the tyranny of the East. Alexander, subsequently, was compassionate in respect of Egyptian culture and respected it. Once conquering the Persians he travelled to the Temple of the Oracle of Amun in the Western Desert and asked for the approval of Amun, which was given.
The culture of Egypt flourished, and even became really popular across Greece in what we Egyptologists call 'Egyptomania", a phase in which Greece would produce wares in Egyptian styles, or even opt for Egyptian style burials etc. Alexander also erected Alexandria in the Western Delta, which became the centre for scholarly progression across the whole of the Med. I'd say "we know what happens next" but honestly... we don't.
The source comes from my own knowledge (I am an Egyptologist, lthough I do primarily work with much earlier periods, and this is only Reddit... remember, dont believe everyone based on just their word), but I will give one or two further readings for if you wish to learn more!
https://www.jstor.org/stable/20546817?seq=1 - This is a background reading, piecing together life in Egypt as a whole in the Macedonian era
https://www.jstor.org/stable/3853895?seq=1 - READ WITH CAUTION... very dated, but some important points
https://www.jstor.org/stable/643071?seq=1 - some stuff about Alexandria
https://www.jstor.org/stable/4436642?seq=1 - This I feel is the most relevant to your question, its about divination in the Ptolemaic era, but covers Alexanderos too.
Hope I've answered your question.