with obsessive characters, it's all about noticing/cataloguing even the minor details of what/who they are obsessed with & desiring to understand them. considering meaning behind every action. constant interest, curiosity, and observation. fixated on everything they do. paired with devotion, they get immersed in someone that they can't live without them.

Write it as seeing a person as a question/ mystery and stopping at nothing to figure them out even if the knowledge might hurt you and change you. Even if by understanding you will come away scathed.

Even if you might not like what you find, you remain devoted to their mystery until you understand them on such a level that they become you and you become them, having learned them like your face in a mirror.

Such study of a person or thing can lead to desire, to poning and yearning for any of their attention to be fixed on the character. Or it can lead to the being character on edge and hyper aware of them, now that the character has unwittingly trained themselves to be so attentive and fixated on them.

Possessiveness comes in when the character wants to be that cause and the focus of all they do. Want them to look at you always, and no one else. Doesn't want anyone to touch them. Wants them to belong to them alone. To be theirs completely. Wants to protect them always.

An obsessive mind consumes things that catch it's interest until the subject is all the character can think about. The subject becomes something of a safe place