I have a 10-year-old son and I cares about him. So, when I get home from work I often ask him to tell me what has happened at school.
He often call this "report".
Is it appropriate to use "report" for this situation? Would it be too formal?
There is no problem with using "report"
One of the reasons for speaking is to communicate. Since both you understand the word "report" in this context, you are communicating. Lots of families will use words in specific ways; it's normal.
If your son is using "report" then I don't see how it could be "too formal", since he has, I assume, developed this piece of languge naturally.