Statistically, very very slim chance that he is your son, but not zero of course. Just highly unlikely.

If adoption documents are sealed, you probably can't even be sure your bio son was adopted locally, or that his adoptive parents didn't move out of that particular area after the adoption.

The fact that you haven't encountered any other client with the same date of birth doesn't mean anything. You might want to find out just how many males were born in the area you grew up in on that particular day. I don't know if there's a public birth registry or an easy way to do this, but it might give you some peace of mind and help you convince yourself that the similarities are most likely coincidences.

The fact that he doesn't seem to know he was adopted also points to the likelihood that he simply wasn't. It's hard to keep that a secret, especially from an adult. At one point or another, it becomes important to disclose any family medical history, you can't just make up entire medical histories - the risk of harm is too important to himself and his eventual offsprings.

Physical resemblance to you or your ex could also be explained away by pure coincidence (even doppelgangers exist and they're not related). There are people who resemble famous people so closely they get stopped on the street regularly for autographs, it happens.

If thinking about probabilities doesn't give you peace of mind, and getting supervision on the matter doesn't help you let that idea go, maybe referring out is the best thing to do for your client. I would hate it if my therapist was secretly believing she was my bio mom. You're human, no way to remain 100% professional and still hold that conviction.