What's the difference between these sentences?
- I think you can't pick the name.
- I think you don't get to pick the name.
There is nothing special about "pick" in this context.
The idiomatic phrase "get to do" means "to have the opportunity to do"
I got to ride a horse when I was at the camp.
It has some sense of "be able and allowed to do something". So the meaning is close to "can", but not exactly the same.
"You don't get to pick the name" means that you aren't allowed to pick the name. "You can't pick the name" could mean that you aren't allowed, or it could mean "you are unable to pick the name".