If your boyfriend of one month with whom you’re sleeping with says he would tell someone he has a woman he is talking to rather than a girlfriend, or that he has someone special in his life, I’d be insulted if I were you. Big time.
Either he doesn’t want anyone to know he’s with you to keep his options open, or he’s trying to make you feel badly about yourself. Both of which are rotten. I mean, even if he wants to keep his options open, he could have been a lot more tactful and told you that he’d tell anyone who asked if he has a woman that it’s none of their business or that’s private, or even a coy, maybe, would be better than what he actually said.
I’d see this as a big flashing yellow light, and proceed with caution.