ANSWER - I assume you have a glazed porcelain ceramic tile with an inkjet wood design.  Those types of tiles are normally very chemical resistant, so it seems unlikely that the glazed surface has etched due to the acidity of the urine.  Although these inkjet tiles are relatively new on the market, so maybe they some of these glazes are softer and acid sensitive in certain cases?

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If the stains are etched conditions then it isn't something you can remove because the structure of the glaze at those spots have been altered by being subjected to the acidic urine.

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There is a chance that maybe it is somewhat of a calcium-like stain that is removable.  You can try cleaning the floor by cleaning it with a detergent cleaner using a bristle brush on an extended handle and not a mop that most of the time just moves the surface dirt around.  Make sure the brush doesn't scratch the surface, which under normal glazed porcelain tile conditions it shouldn't.  Use a wet and dry vacuum to pick up the dirty water and then rinse the floor with clean water and pick it up with the vacuum.  You can then let it air dry.  When water doesn't bead up from the surface of the tile our grout to some degree, it is time to reseal.

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The key to maintaining tile floors is to properly seal it, and properly maintain it.

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There are penetrating types of sealers such as Miracle 511 Impregnator that works well over glazed surfaces.  The sealer doesn't make the tile or grout waterproof or stain proof, but it will make them resistant to moisture and staining.  The sealer will make the tile easier to maintain if the sealer is maintained.  Normally you have to reapply every 6 months or so depending on the degree of traffic and wear and tear that it is subjected to.  The spills need to be cleaned timely and properly to prevent staining.  e.g. can't use water to clean up oil stains.  Must use a detergent; thus use the appropriate cleaner for the respective type of spill or stain.  When cleaning up urine make sure you rinse it well so there are no residuals left to affect the surface.  Here is a list of manufacturers who make tile cleaners and sealers with links to their website.

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Key to applying sealer is to first make sure the floor and grout is perfectly clean and dry or you will trap in stains and moisture.  When you apply the sealer make sure that within 5 minutes you wipe the surface dry with a lint free cloth. Otherwise the sealer can leave a sticky residue that can cause the floor to pick up dirt and can peel off.  Good luck.