I don’t know whether or not it is possible to have arthritis in the chest, but in referrence to the ‘you have no joints there, so you can’t hav a joint disease there’ argument, you DO have joints there. A joint is any place where two or more bones meet. You even have joints in your skull, actually.
Arthirtis pain differs for everyone, but most often it is described as aching, burning, crushing, etc. (You should go to the Arthritis Foundation web site; they can give you a lot of information.)
There are other, similar conditions that can affect that part of the body and feel like arthritis. The first that comes to mind is fibromyalgia; one of the “tender point” pairs used for diagnosis is located in the chest near the collar bone.
If it is arthritis, your doctor can tell with a blood test; if it is something else, they’ll probably need a few more tests to know what it is.