Here's a frame grab from the 'Burke's Law' episode "Who Killed April?" in which the detectives are asking for the location of the manager for the fast food restaurant. This was the only scene for that car-hop waitress and she was played by Rue MacClanahan of 'The Golden Girls' fame.
She only had the one line but never identified herself. So I'm going to make the claim that one day she would become the powerful gossip columnist Verity Chandler, as seen in the 'Columbo' episode "Ashes To Ashes". Just about everybody in show business had to start somewhere, and for Verity Chandler the job of a car-hop waitress would have been as good as any to keep the bills paid. Who knows? Maybe somebody of importance - a movie producer or a studio executive (May they be nibbled to death by ducks!) - might have pulled up for a burger and shake. If she played her cards right, (and Verity Chandler did seem like a real operator in that 'Columbo' episode), she might have then finagled a job for herself... one way or t'other. Nudge nudge, wink wink......
Eventually, after years at the top in her exalted position of knowing the secrets of everybody in Tinseltown, Verity Chandler ran afoul of the "funeral director to the stars", Erik Prince, for the graveyard dirt she had on him.
I suppose someone with a high-def method of checking the details in her 'Burke's Law' moment of glory could figure out what her name was on her uniform. But it wouldn't matter if it was different from Verity Chandler. Once she was on her way up the ladder of success, she could have changed her name like so many do in show business.
And if even "Chandler" wasn't her real name, where did she get it? I think we can look within 'Columbo' for that answer - from Nora Chandler, who would have been a big star back in the late 40's into the 1960's. She could have been a major influence and a favorite star of Verity's. (Nora Chandler had a secret that not even Verity Chandler was able to figure out.....) And as for "Verity", maybe there was a televersion of the producer Verity Lambert, best known in the real world for shepherding 'Doctor Who' in the beginning. Or maybe Verity was a character in some fictional bodice-ripper novel found only in Toobworld. Perhaps in some classic murder mystery novel by Dame Margot Woodhouse who would go on to write "Death Rinses Out A Few Things" near the end of her career (from 'Dream On'), or "The Rooster Crowed at Midnight" by Abigail Porterfield* (as seen in 'M*A*S*H').
Or better yet, Verity could have even been a movie character played by Nora Chandler, keeping that riff going a bit longer.....
At any rate, this "Game Of The Name" theory would place 'Columbo' and 'Burke's Law' into each other's orbit - which they should be since both of them are attached to the Los Angeles Homicide Division in Toobworld......
'Murder, She Wrote'
* Abigail Porterfield would also go on to inspire Jessica Fletcher (of 'Murder, She Wrote') in her literary career, but under her proper title of Lady Abigail Austin (seen in the episode "The Grand Old Lady".....) And there's a theoretical link in itself!