It's likely to be a different Pharaoh, but nevertheless, the name "Moses" was a big deal in Egypt. When we say Ramses or Tutmose or Amenmose then that is the same name, a theophoric name meaning "child of [God]", the omitted God's name most likely the Nile, from which Moses was drawn out. Only Pharaohs were children of gods. It's a royal name.
People didn't just walk around saying "My name is Moses". Nor would anyone believe you if you wandered in from the desert and claimed this name but weren't able to provide evidence of your royal status. Moreover the story of a member of the royal household killing an Egyptian and fleeing to live in the desert is something that many people would remember and tell their children.
So then we have a few options:
- Was there a statute of limitations that expired?
- Given that Moses appeared with the power of God, perhaps Pharoah decided to deal with him for the sake of diplomacy rather than enforce the law?
- Mass amnesia
#2 and #3 are unlikely because Moses would first need to get an audience with Pharaoh before he could perform any miracles. And you don't just walk into Egypt as a no-name goat herder and speak to Pharaoh. Moses most likely had to use his name and royal titles to gain entrance, which suggests people recognized who he was. It's then hard to argue that they forgot why he fled. Moreover, Pharaoh didn't appear as one who would respect the Hebrews enough to give them some kind of diplomatic status.
The jews also likely recognized who he was and, as the most prominent (highest ranking) jew in Egypt, they would at least listen to him, but as a random goat herder talking about seeing a burning bush, they would be less likely to listen.
So it remains a mystery, except we can say this was the power of God at work.
According to Christian Gedge's chronology (which I think are the most accurate chronologies that reconstruct the pre-metonic Hebrew calendar), 1396-1395 was a jubilee period (jubilees were periods of forgiveness and restoration that were in effect every 49 years from month 7 of the 49th year to the end of month 6 of the 50th year), so with an exodus of 1444 when Moses is 80, we have another Jubilee from month 7 of 1445 to month 6 of 1444. Thus if Moses returned to Egypt during that time, he would be returning in a jubilee year, and perhaps this marks why the murder was forgiven or at least overlooked.
Note that I am not saying that Egyptians honored jubilees, or that anyone was even aware of them at that time, but that God honored it, and arranged it so that Moses' murder would not be held against him in that year. It certainly seems like a curious coincidence that this is the year in which Moses returns from hiding and is not prosecuted, and is also 49 years before the next jubilee period.