I'm uncomfortable/uncertain about being friends with a 13yo
Why makes you feel uncomfortable about that?
Overall I just feel really different from kids that age.
That's probably because of the age difference there, assuming you're an adult.  That's why adults shouldn't be friends with children.  You can be friendly to them, of course!
How am I supposed to know that the child is the child of a friend? I'm going to be more careful.
I'm glad you plan on being more careful. Like I said you should always be friendly with children, but becoming friends with them outside of a family setting is a bit odd.