Fact 1: The vet records for your cat say she is neutered.
Fact 2: She is exhibiting the behavior of a cat in heat: loud (I mean, loud) yowling, peeing outside the box, poking her butt in the air (apparently, all of this is quite erotic to a tomcat)
1. The vet is running a scam where he charges the shelter for neutering the cat but doesn't really do it
2. The vet does a bad job of neutering, not getting out all the ovary
3. Your cat is not the cat on the record
This, of course, is one of Gary Larson's brilliant Far Side cartoons.
And with the miracle of technology and microchips, we find that Shirley is not Cat #4a095d0859 but is 4b0e454834.
In the private sector, people lose jobs over things like this.
In the public, non-profit, crazy cat ladies sector, I guess we're just glad someone will do this job.