There's a lot of variables involved, like your state, jurisdiction, and zealousness of your local prosecutor, but yes. For crimes as minor as public urination, it is possible to be charged as a sex offender. My county's DA has made a career for himself as being "tough on crime" in regard to sex offenders. Many politicians have done the same. A prison near me had to split itself in half, general pop one side, chomo ward on the other side, because there were so many inmates classified as sex offenders. It's an effect of widening the definition so drastically in recent years.
And remember, a lot of people who plead guilty to a charge can be innocent of that specific crime, but they agree to plead to it as part of a deal. Very often, if you plead out and admit what they tell you to admit, then you can get off with probation. But if you fight it, you're not going to win anyway, and you'll end up in prison. People will sign anything to stay out of prison. I probably would, too.