It's also not a club (despite the FC in the name, it's a franchise).
Why not both? The Phoenix fit both definitions.
To its most important stakeholders - its fans - it's a football club. It should remember that in its marketing.
Also, a franchise implies an organised, consistent process for operations and marketing - from the franchise owner, FFA. Er...
Franchise is a model of ownership used commonly in Australian sports as well as in the USA and other markets.
In my daily life as a football tragic, I do not care for the difference - to me it is The Club, even if it is a "franchise".