What? No. When you are 0, it is your first year. When you are 21, you have begun your 22nd year. In the US you are legal to drink in your 22nd year of life.
You are correct that nobody says "22nd year" in this context, but nobody says "21st year" either. The former is awkward but the latter is just incorrect.
You are correct that nobody says "22nd year" in this context, but nobody says "21st year" either. The former is awkward but the latter is just incorrect.