I suspect you're over thinking the scenario. Consider what happens if you get an injection at your doctor's office or if you have diabetes and you're giving yourself a shot. Regardless of whether there's fur (and with humans, there may be very fine hairs), there's no incidence of contamination. You don't get your family sick. I've been on this site for a very long time and not heard of anyone having a problem with a pet-borne illness. Likewise, I've work at a medical center for decades -- before it was standard practice to glove for giving shots and never had a medical professional become contaminated unless they stuck themselves with a needle. We have had members inadvertently stick themselves after having given their cat an insulin shot and not infected themselves. Remember, for a pet to infect you, the animal would need to be ill.
Also, consider that your cat cleans him/herself quite thoroughly, as do dogs. They will lick from one end to the other and then snuggle up to give you a kiss. We don't get sick.