I have family from the old country with two names and none of them are bothered when someone uses their adopted name because it's easier or just what they know. They picked names they liked for that reason. Yes, I asked them.
She should be called "Kara" or "Pee Gee" in the field if not "Power Girl," same as Clark does when he's operating as Superman. I agree. It's bad writing when they just call her Karen. That said, PG only sometimes cares about her secret identity. There are large swaths of time where she's more like Diana and sometimes Dinah-- PG all the time. When they're calling her Karen as PG, I assume it's one of those times where reality has been reset and "Karen Starr" isn't really a secret anymore. Is that a cop-out? Sure, but DC hasn't paid a lot of attention to Karen's private life.
In her own words:
PG.jpg
The whole point of being PG is to distance herself from anyone's legacy and build something entirely her own, something even Dick Grayson didn't do when he chose to honor his surrogate uncle Clark in adopting the Nightwing moniker when he took his own path.