How do you say this in English (US)? cat, that always bothers and disturbs you. Can I say : He is obsessive cat ?

See a translation

Annoying would probably be best. Or, if you mean to say he often demands attention from you, you could say high-maintenance. If the cat bothers you because it wants your love and affection, you could say it is a needy cat.

annoying = the cat bothers you, or does stupid things that make you mad sometimes.

high maintenance = the cat takes a lot effort, and it takes up a lot of your time to care for it

needy = the cat always wants you to pet it or give it attention because it loves you so much

Obsessive doesn’t really work here unless you are mentioning what the cat obsesses over. For example:
“My cat spends all day staring at my fish tank. It’s obsessive.”