Don’t confuse common descent with common design.
Our DNA would be expected to be very similar because we look similar and live in virtually identical environments and eat basically the same food.
Would you consider it evidence of intelligent design or creation if genetically we were wildly different?
Eating basically the same food makes your DNA similar to other creatures? My dog will eat pizza all day long. How long before that changes him into a hominid?
According to your “like should be like” hypothesis a chimpanzee and a gorilla should be more similar in DNA than either is to a human, but that is not the case because a human and a chimp are more similar to each other than either is to a gorilla. Old world and New world vultures should be more similar to each other than either is to hawks or cranes, but they are not.
My dog also lives in the exact same environment as me.
Our DNA must be becoming more identical, right?
What you eat has no effect on your DNA other than to keep it functional and allow it to reproduce.
However, organisms living in similar environments may be pressured, over time, to make similar adaptations.
This concept of similar environments producing similar creatures confused early taxonomers. It confuses some of us still ... "The marsupial wolf is not a wolf? But it looks like a wolf!"
The clue is in the word marsupial.