Is my wood cabinet exposing asbestos?
Is it possible that we are releasing fibers?

Anything is possible, but it is highly unlikely that incidental contact would release fibers. It is also highly unlikely that there is any appreciable amount of asbestos there (if any at all).
There is a ZERO percent chance, assuming there is asbestos there, that the scenario you are describing would in any way release enough asbestos fibers for you to worry about a health concern.