I'd second what others have said here - I think it would be a useful thing to share with your therapist.
And, just to add, I do understand that whole "getting on a mission to know stuff" thing. For you, you became focused on his surname. For me, I got on quite a mission to get certainty around my timeline/age. I succeeded in my mission too - I found out certain dates and they weren't what I expected. I had had all my timeline wrong. While, on the one hand, it was satisfying to finally have certainty around the information I had sought out, on the other hand, the discovery was unsettling. And I had a very similar sense of "so, what now?" as you're describing here.
It was an odd experience. I felt quite lost with that new info. And I'm glad that I talked it through with my therapist.
Hope your session goes well today!