Is professorship (assistant/associate and full) a title or a position?
In the US there are no legal rules. I think that few people would use Associate Professor as a title, but rather as a description of the post they currently hold. They would probably use Professor as a title, however. It is a descriptive title even before elevation to Full Professor. Students normally call them "prof" in many places. But the "title" is descriptive, not formal. It is a term of respect when used in this way. 

However, if a person leaves academia for some other profession, it is unlikely that they would use professor and variations as a title. They might describe themselves as "former professor". 

After retirement you might be "professor emeritus", again a description. On your tombstone you might be "Dr. Stone, Professor of English", the final description. 

Other places are more formal about such things, but in the US, not so much. 



I am actually "Professor Emeritus", which I interpret to mean that I didn't unforgivably disgrace myself before I retired.