I have not dealt with this problem and ceramic tile but have experience with other floorings (vinyl, carpet, hardwood). I think you probably do need to replace the flooring and take a long look at any sub flooring. Anytime the urine gets all the way to subfloor you have to remove your flooring at least that far to let it dry and then seal it with something like Kilz before putting new flooring down.
My grandma had a condo with carpet and an incontinent cat for years. When the cat passed away, she had the carpet replaced. They took out both the carpet and pad but installed new carpet on the same day without addressing the soaked sub flooring. Obviously, that didn't eliminate the cat pee odor.
At another house where we had some peeing problems, we replaced carpet with hardwood floors. We cleaned the stains on the sub flooring the best we could, then let them dry for a few days before sealing them with Kilz. This approach was successful.
Couple more products i really like after caring for animals in trouble for many years, is citra-solv ( i buy the gallon on internet) and also Nature's Miracle. Both good for odor control. Never use anything you could not use next to a bird. No poisons.
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