I think this has been brought up before. Heres some possibilities:
1.If he was a captain he probably wouldn't be out working on cases anymore, he'd just send guys out and they would report to him. Its in Columbos blood, he can't walk away from it. Stubbornly he would refuse the promotion.
2.Or maybe they don't want to promote him because hes too darn good at what he does; who would replace him?
When watching Columbo you pretty much have to throw any sense of realism out the window.
Agreed. With all the celebrities he has arrested, he would be the most renowned cop in the world. He even got publicity, whether he wanted it or not, in Columbo Cries Wolf. Any one of these arrests would have made him famous.