Both Mesopotamia and Egypt worshiped their gods through service to the king and his dynasty. He was their earthly connection to their gods. Both cultures produced carved scenes of their kings performing rituals and giving offerings to the gods. They both also used size to depict the importance of rulers and gods, as in the Palette of King Narmer and the Stele of Naram-sin. They also share some mythology — the serpopards on the Palette also show up on a Mesopotamian cylinder seal, but not in such nice detail.
Although both cultures came up with a writing system, the Egyptian written language was more elaborately decorative. Mesopotamians came up with clay tablets, and Egyptians created paper that could be carried and stored more easily. Beauty seemed to be more important to Egyptians when it came to visual communication, so even the Hieratic script they developed was full of curls and flourishes. Cuniform was more limited, but so was the ability to make paper in Mesopotamia.
Egyptian kings would donate art and ritual objects to the temples, but Mesopotamian rulers would have votives of themselves made to stand in supplication in their temples while still alive. Egyptians would have statues of their dead ancestors standing in their family temples. I also noticed that images of people in tombs or temples are barefoot in both Egypt and Mesopotamia, showing that they are on holy ground.
Both cultures had the idea that the statues and scenes of their rulers had an impact far after the death of the person being depicted. Egyptian and Mesopotamian art was vandalized specifically to remove the eyes or ears of the rulers, to take away their power after death.
Mesopotamians and Egyptians both came up with pottery, with decorative scenes of animals and repetitious symbols. Their pottery started out rough, then became delicate as they developed the pottery wheel. I assume their pottery decoration became finer from the wheel as well, in being able to paint lines for registers swift and even.
Just like the Mesopotamians, Egyptians depicted their rulers as ideals of godly beauty rather than in realistic portraiture. For example, Queen Nefertiti’s bust looks like a Prada model, but digital mapping and 3D reconstruction of her mummy’s face show someone we wouldn’t recognize as the queen of the Nile. Statues of rulers were almost completely symmetrical, strongly facing forward with stiffly posed limbs to impart a sense of mastery. Interesting that later rulers had themselves portrayed as wise and older.
My favorite similarity between the two cultures is that they weren’t shy to scavenge brick from the building projects of previous kings. It’s a pain to the kings in the afterlife, and it’s a pain to modern anthropologists who are trying to piece history back together. First rate jerkery!