What do you call someone who is constantly obssessed with spending time with you and constantly wanting to socialise?

"Clingy" or "needy" (as noted previously) work. I would use "clinging" as a slightly more formal construction. English offers a wide range of metaphors for this. A "limpet" comes to mind, although one could be much more pejorative and colorful.

In British English there's a verb to tag along, which means to join a group that's going somewhere. Sometimes this is used to denote an effort that is unwelcomed by said group. It's defined as such in the Cambridge Dictionary:

tag along: to go ​somewhere with a ​person or ​group, usually when they have not ​asked you to go with them: I don't ​know her, she just ​tagged along with us

However, this term has started to be used as a noun, i.e. an unwelcome individual is a tag-along.

I can't find a reference book to back up my claim that the term's used in this way, but if you do a search on Google for 'he's a tag along', you'll find many examples in literature. Here's a good one from The Facts of Life (2010) by Graham Joyce:

'He's nice enough but as far as I can see he's got no work either. He's a tag-along, isn't he? Well he's decent enough, but you can't tag along with a tag-along.