Is caring a lot about your birthday normal when you're an adult?

Much depends on your education, family habits, values, and being able to think independently.
In my opinion B'day is rather arbitrary. If I want t to offer somebody something, why should I wait the B'day and if I have no inspiration on that day it would be silly to feel pressurized by it.

your birthday only comes once a year; your life matters, so your birthday does, too.

If ur a kid at heart yes