I was a green card holder from the age of 5 until about 33.
That is correct. You ARE REQUIRED by LAW to have your green card on your person at all times, starting at the age of 18.
You used to have to also report your current address every year, but that is no longer required.
When I was young I remember an ad on TV reminding all resident aliens in the US to report to the Post Office once a year.