I haven't had the option for a collar either. Book and sofa, yes, but not collar. Is there something that trigger that?
There are three collars: Blue, Pink and Black. I'll try not to spoil the actual story, but I will spoil quite a few game mechanics.
The Black one is the easiest - just let your Naughtiness go through the roof* and eventually the game will let you give Lith the collar.
The Blue collar is also easy - just talk a lot and try to keep things as platonic as possible**. The key here is to listen to Lith's backstory. After that reach 50 Niceness and ask him how he feels. This unlocks the option to give him the Blue collar.
And for the Pink one you'll have to make love with Lith without raising your Naughtiness above 20-25. You can ask Lith what he likes until he admits his fondness of snuggling; you can Feel him up a few times to make him open up about his nudity; you can offer yourself to him when he's aroused. After that, as with the Blue collar route, reach 50 Niceness and ask him how he feels.
Edit: you also might want to raise Lith's arousal and then ask how he feels.
*Note that once you do any GRAB-bing (aside from kissing) and, subsequently, let your naughtiness go above 20-25,
the game will permanently lock you out of Blue and Pink collar routes.
**Technically it's possible to unlock the options to give Lith Blue and Pink collars in the same playthrough. You'll just have to be extra careful with the Naughtiness stat. Don't let it go above 10 and you're good. But you're only able to give one, so choose wisely.
Do note that to get the pink collar, I think you need to have breasts. It's a thing related to Lith's backstory, as explained in the stories the creator has written about his past.
Anatomy is irrelevant when it comes to being able to get the collars.