A couple of thoughts.
First. This may just be the same urinary tract infection. The only way to be sure that a UTI is gone is to perform a sterile urine collection (cystocentesis) 10 - 14 days after completion of the antibiotic course and to have a negative urine culture test results. (and this still would not be 100% sure). With this said, for a first time infection, this is rarely done. I usually do exactly as your vet did - UTI signs, urinalysis that supported infection, 2-3 week course of antibioitics and 'see how it goes'.
When it 'comes back' - we are not sure if it is a new infection or if the first infection was just suppressed clinically for a period of time from the antibiotic course. So then I would do, again as your vet did, repeat urinalysis and add on blood work, radiographs and urine culture. When the test results come back, then I would adjust the antibiotics as needed and repeat a urine culture 10-14 days after completion of the medication.
If at that time the culture comes back negative, then I would monitor for recurrence. If the UTI recurred, then we would have a pet that is having recurrent infections. For females, since their urethra is shorter and wider than males, often times they get ascending urinary tract infections (bacteria that moves from the vulva, up the urethra and into the bladder). Individual susceptibility and anatomy are usually the cause of this. Obesity can help predispose pets to UTIs. Occasionally it is environmental if they get a lot of dirt/debris/feces on or around the vulva.
If the culture comes back positive, then it is important to note if it is the same bacteria or a different bacteria. Some UTIs need a prolonged course of antibiotics and some will switch from one bacteria to another due to suppression by the antibiotic and resistance of the other bacteria.
So at this point, I do not think there is any reason to panic. There may be a simple solution to this. Lets monitor the patient and see how this culture and the follow up culture turn out.
Hope this helps.