This is one of those YMMV factors, though indeed most hotels do have an official "21 and older" policy.

However, I traveled extensively through my younger years and had Hyatt Diamond Status, Marriott Platinum Status, and SPG Platinum Status all before I turned 21. Never had a problem with ANY hotel denying me the ability to check-in, even before attaining elite status. I even called a hotel once who had it prominently displayed in several places on their page, and the manager confirmed that since I was Platinum it would not be a problem.