Sometimes "have" is the helping verb and the main verb. I "have had it" is in the Present Perfect tense to describe an action (in this case possession of the watch) that started in the past and has continued until the present time. Don't stress over this too much because the question is extremely poorly written. As you noticed, B could also be grammatically correct since "keeping" something can also be an action with a duration (although it makes less sense here than talking about "having" it for two months. In addition, "after I worked here" is not a well-written description of when the action started. It should be something more like "after I started working here."

All the answers are wrong. The sentence gives no information about how long the person owned or has owned the watch or whether he now owns it or not.

'have kept' is potentially grammatically correct, but not correct in the context, and it's fairly uncommon in any case. The use of kept here suggests he has it without having the right to it, so for example: "after stealing the car, I have kept it for two months, but I still plan to return it at some stage".

I agree that the question is poorly written and doesn't sound to me like it was written by a native speaker. "You have had it for 2 months" means you have possessed it during that time. "You have kept it for 2 months" means you actively did not give up or get rid of the item. It implies more intention. I might have a watch, but I might be keeping my grandfather's watch because it's meaningful to me. Or I might keep a cheap watch (i.e., decide not to get rid of it) even though it's ugly because I don't want to spend the money on a better one.