I've noticed the announcers starting to say "city" when referring to our club. Not all the time but definitely enough to notice.
I for one don't like referring to the team as "city" because I think it just emphasizes the link to CFG/MCFC and makes the team feel less unique to me. And for me, "blues" is more of a descriptor than a nickname... it's a color (and there are TONS of "blue" team in MLS). Just my opinion.
I like the Pigeons thing because I think it's fun/unique and fan created. There will never be unanimity about any team name... so people will run with what they run with. More so I just like the idea of having a nickname... ANY nickname... it's tradition in American sports and it gives the team character/flavor. There are a lot of Squirrels in NYC too, just sayin...