I would hope not, considering many major cities are serviced by multiple major airports.
NYC - JFK LGA EGW
PARIS - CDG ORY
London - LGW LHR
Tokyo - NRT HND
SF - SFO SJC
If someone starts refering to a city by the airport code I'd be completely lost because most airports and their codes don't reflect the name of the city they service.
SJC doesn't really serve SF. You can say OAK sorta does, but not really. It's pretty clearly SFO. Also, the BART line to SF is called SFO (used to be called whatever terminating city until they realized that's stupid).
From my experience doing so is a good way of coming across as a douche tech-bro (“yeah, I’m so cool I use code names”). Particularly when the comprehensible way would be to write SF rather than SFO.
We have a datacenter there, so I'd totally know exactly what city you meant by that. I wouldn't say we always call Chicago ORD, but it definitely happens.
MTV, SFO, MPK and all of this lingo is not familiar to all