The series was originally executed as following the exploits of the crime reporter wife of Lt. Columbo, but when the series couldn't capitalize on the popular earlier series, producers changed Mrs. Columbo's name to "Callahan" and soon, all references to the fictional cop were removed.
If you watch the original airing of the pilot at the very end when they are performing on stage supposedly the lieutenant shows up in the back of the theatre. You can see a back shot of a man in a rumpled raincoat with curly hair.
While the show aired under many titles, it was officially renamed "Kate Loves a Mystery" in the second season, and all references to Columbo (1971) were removed. The storyline explained it by having Kate divorce Columbo. It's not clear whether season two is a continuation or a separate series.
Mrs. Columbo's name in the original series was rarely referred to, but it was Rose. Not Kate. Perhaps her full name was Katherine Rose Columbo and the Lieutenant merely preferred to call her Rose.
Peter Falk was in his early 50s at the time. Kate Mulgrew, in her mid-20s, was considered far too young to be cast as the wife of Columbo.