Inform the search committee of another university about your current campus interview I applied for some places and got a campus interview from one of them. My second application, however, has remained silent and I have not heard any news yet. I email the staff who is in charge of this position (answering email and uploading the missing documents, ...). She said that "We are still in the selection process. We'll notify you if we need further information from you." I think this is just a very general response without any clue and maybe they already have selected the finalists!

I was wondering if it is wise to let the search committee that I already have a campus interview and want to get an update on my application? Because I like this institute instead of the one that has invited me.
I suspect that what you suggest is risky. It might bump them in either direction. If they are interested in you they will let you know, but probably on their own schedule. If you try to force the issue, it may be easier for them to say good-bye unless they really really want you. 

Some places have fairly strict rules about things and may not be able to say more than they have. That would result in a general answer, of course. 

I'd suggest you don't press too hard. Go to the other interview, evaluate what you hear there. If it sounds good, you can always ask for a delayed acceptance to see what happens at your preferred place. It might be acceptable or not, in which case you need to make a decision. 

I would also guess that they aren't so naive as to think that you aren't exploring other options. It is certainly assumed that you are.