No, cities can have cathedrals (although not necessarily Catholic) to be cities but other things like good universities (e.g. Cambridge) can also allow a city to be designated as such. Factors like...
Albi has a brick cathedral, Amiens has a cathedral listed in the World heritage sites, Clermont-Ferrand has a cathedral built in black volcanic stone, Bourges, Chartres, Reims, Beauvais, Strasbourg,...