In fact, the more I think about your phrasing of it, the more I think that, in his teenage years, it was more purely possession than it was for the child or the grown man. The fact that he went to Voldemort to have her spared when James and Harry were killed, and was still willing to trade them for her when he went to Dumbledore, and thought that'd all work out for him... that's very purely possession of an object by that point, she's thoroughly objectified in his head. The child loved her, the grown man loved her, but 15-21-year-old Snape wasn't capable of it, he did just want to possess her. She became the tool that Dumbledore was able to use to teach him that once, as a young child, Snape had been capable of much more humanity, in part because it was so clear that something had gone very wrong in the way he cared for her.