I think you mean "veterinary" disinfectant? At least I've never heard of anyone having to disinfect their veterinarian? Mine have always been pretty clean...

But seriously , I've used F10, Nolvasan, & generic chlorhexidine for many years. They are designed to kill 'germs' & many germs are harmless, even necessary to
life, so none of these products belong on your skin (& certainly not in anyone's eyes) for any length of time. When properly diluted for wiping cages, I don't feel the
need to wear gloves, as very little contacts anything other than my paper towels, but you should avoid saturating your skin. And they don't belong in or on the hands
of children, goes without saying (I hope). If you read the precautions on the label for any of these excellent products, they also pose a risk to wildlife, so you want
to take care how you dispose of surplus. Basically, used as directed (& diluted as directed) all are very safe, & need no rinsing from the cages...but please don't bathe
in any of them, & you'll be fine. (FYI, chlorhexidine is used in mouthwashes...but humans are much bigger than most of our pets, & mouthwash has other ingredients,
& isn't designed to be swallowed anyway.)