In order to join the silver elite, one has to know what Sephiroth has named his attack, sword, and that he is left handed. The club mentions Sephiroth's shampoo secrets, his hang outs in the training room with his friends, and offers rare photos of him. First of all, why would Hojo care about such insignificant things like his prized specimen's social life and bathing habits? Wouldn't a normal father boast about his son in other ways? That's more than "that's my boy!" And those things are certainly not connected to his value as a specimen because those things are not scientific or about his power. That means that Hojo actually keeps photos of Sephiroth in his possession, and in order to know his shampoo habits, must either have access to his financial purchases and sees that he buys multiple bottles of both shampoo and conditioner multiple times a week, or has cameras set up in his bathroom and watches him every time he bathes (which I don't think Hojo has time for). And the part about Sephiroth playing the dumb apple game with Angeal and Genesis in the soldier training room? Who even has access to place to know that? Hojo. He must have watched old recordings of their interactions. Is such a silly game related to scientific observation? No. To me it seems creepy and sweet, to think Hojo was interested in the little, personal things about Sephiroth. He certainly did not pay that kind of attention to Red or Aerith.
It's typical for celebrities to have stalkers, I see stories on the news sometimes. They have their privacy invaded with the intent to harm or scare the victim. Hojo certainly didn't seem to have that in mind. I think that because Specimen S was his child, he felt attached to him more than he did to his various other specimens. He did not want closeness or intimacy with anyone because he was afraid of being vulnerable and wanted to be the scientist who studies. I think he felt proud that this awesome man was his child. This tall, muscular, decent looking super powerful warrior and he was the typical loner eccentric quiet quirky scrawny four eyed awkward geeky misfit. He probably got all giddy inside thinking " I can't believe I fathered this man!" I think it's almost affectionate in a weird creepy way, but yes it is an invasion of privacy and his ego might be involved. Who knows what other personal things he knew about Sephiroth but never talked about. After all, Hojo is the only one who saw Sephiroth before he was perfect. He held the crying infant, witnessed Sephiroth learn to walk and talk, etc.