Definitely not City of Manchester. Stockport is one of the other 9 Metropolitan Boroughs that make up the Greater Manchester conurbation.
Being based on an important town, I’d put it in the third tier with people mostly identifying as “Stopfordians” (i. e. from Stockport). I’m sure a fair number would consider themselves Mancs or even “dual nationality”. Particularly City fans!
I could be wrong, particularly as I’ve been away from the area for a number of years.
It’s often the case that someone will say they are from Manchester, or “near Manchester” when they first meet someone from another part of the country or overseas. If they find the person knows the area, they will go on to be more specific.