I think that many doctors make chit chat to make their patients more comfortable. If you have a vibe that wonders if he is flirting, then don't go back. Find a different one. It's a breach of trust and he could lose his license. But in general, chatting and sharing some birth details is not strange in this environment.
Are you saying you like it or don't like it? If it creeps you out, ask for a recommendation to another chiropractor and go to that one instead and don't bother to explain. If you like it, you might be tempted to encourage it at some level, and he has a professional code of ethics to live up to and could get into trouble for inappropriate behavior with a patient. So, don't go there trying to get into a flirtation or you could cause trouble for someone you really actually like. If you like it and don't mind finding out his intentions, you could talk to him at your next visit and ask for a referral to another chiro, and just tell him you find him personally attractive and would prefer to only have a professional relationship with any treating doctor. Then the ball is in his court. But don't be his patient and start seeing him romantically. When the romance comes in the door (if it does), you go out the office door and don't come back in it.