As a man my perspective here is limited since I haven't been on the receiving end of this, but I think you're absolutely right here:
I think sometimes I feel fear that if I don't say "I have a boyfriend" that the man may become aggressive, especially in situations where alcohol is involved and to be honest this has happened to me. When I really think about that sentence though, the fact that someone can only respect what I am saying if I am another man's "property", instead of just accepting the non-interest is just absolutely unacceptable.
I think men often aren't aware of how threatening they can be to women. I see myself as a nice guy who would never try to hurt or intimidate anyone, but the reality is that I'm physically larger than most women, and to a woman who doesn't know me that alone can be intimidating and scary. Especially since, as far as she knows, I could be a raging misogynist who'll get violent if she turns me down.
This is something men need to take responsibility for. We need to recognize that there are times and places when it's just not appropriate to hit on a woman, and even when the time seems right, we need to exercise an abundance of caution and self-awareness so as not to create a threatening situation.
And above all men need to learn to take a hint. As you say, a woman shouldn't have to provide proof of "ownership" to get a man to back off.