See I think this is interesting, but very different from my own views on the matter.
After all, what is it that makes a person who they are? That’s the real question. One of the most important factors of personality is, of course, memory. I’m not going to try and get overly philosophic or whatever, I have no experience in the areas of memory loss or anything like that. But here are a few observations:
Everyone always acts as if 10 and 10.5 are completely different people. As if 10.5 is a fake, a shadow, an ‘also-and’, a replacement. And this whole thing about “people he never met, things he never did, places he’s never been and a life he’s never lived” Is only reinforcing this. 10.5, if we want to call him that, does NOT have memories of things he never did - he has memories of things he did do - he, himself. I honestly believe that he really is JUST as much of the Doctor as 10 or 11 are.
When 10.5 woke up on the floor of the TARDIS, I imagine he was completely convinced that he was The Doctor - THE Doctor. Because he IS. Then he remembered what happened, he saw Donna, and he felt a few things - things about himself - that he wasn’t used to. He realized his connection with Donna, and his single heart. But before all of that, he is THE DOCTOR. He dosen’t just remember the Doctor’s life, he LIVED it. First hand. All of his adventures, from a child on through 9 regenerations, through the time war, all the time he spent with Rose, when he lost her, with the Master, with Martha, with Donna, all the way to his reunion with Rose. He LIVED that. HE did.
And while I’m add it, I’ll mention this: While there are human aspects to 10.5, he is NOT “half-human”. Because, as we know all too painfully well from Donna’s departure, there has never been a Time Lord - Human metacrisis: Because there can’t be. This almost kills Donna, but it does not bother 10.5. This is because he is far MORE Time Lord than he is human. His regenerative ability and his second heart are the only recorded differences - The only things used up by the other 10th Doctor during his partial regeneration. I don’t want to say exactly how human 10.5 is, but I can assure you it is far less than 50%.
10.5’s love for Rose was real and YES, he did have a choice in it. I don’t think 10.5 did ANYTHING just because his counterpart did. 10.5 loves Rose because he loves Rose. He always has, since they first met in that basement. Because yes, that was him. He didn’t “have” to love Rose any more than 10 or 9 did. He chose to.