People from Manchester are Mancunian or manc.
I don't think Liverpudlians mind the term. The problems come with regional boundaries, for example, Scouses have to be from Merseyside (on the river Mersey), Cockneys within the sound of Bow Bells and Geordies on the river Tyne. They consider people outside the boundary as foreigners, even if they have a similar accent!