Despite using two hangovers to combat the man flu it has now properly kicked in. Anyways onto the vinyl.
From the picture the wood effect floor looks like it is also a vinyl. Heat a paper clip with a lighter and push into an inconspicuous area of the floor. If it melts in its plastic and is therefore vinyl.
As for the slate effect stuff. It may be that a top coat has been applied and this has been affected by the urine. Try and ascertain if there has been a seal applied by the owner or installer previously. If not you may get away with lightly abrading the wear layer with some fine grade steel wool or even a diamond floor pad centre like a kgs flexes, super shine of twister to lighten it.
If yes then you may need to strip the sealer off. A good machine clean with a quality cleaner or stripper, vac off, rinse and apply a finish.
Looking at the scratches on the slate effect stuff I would go with something fairly low sheen like Pioneer Eclipse Floormat so as not to highlight the damage or any that will occur afterwards. There look to be quite a lot of claw marks and they should walk their dog more or file their claws more. A high gloss will mean higher maintenance.
They should also do an audit of any chairs and check the quality of the legs/feet and fit pads where required and make sure they have adequate matting to stop ingress of grit. Sweep regularly and use a flat microfibre mop and a spray of neutral cleaner for ongoing maintenance.
