Vets do know what meds cat can tolerate, but some cats do have allergies and have reactions to certain medications. The vet has no way of knowing that until the pet has a had a reaction. Tylenol is absolutely lethal to cats. People will give their cat Tylenol and then end up in the vets office because the cat is sick and they have no idea why. The minute the blood is drawn, you can tell if Tylenol was administered because the blood is brown and sludgy. It literally looks like a chocolate milkshake. Sometimes it can be reversed...if the cat is brought in in time, but most people wait, thinking the cat will feel better on his own. It's any drug with acetaminophen. It's not recommended for dogs, either, unless a vet specifies the dosage.
My old boy has arthritis and I give him a baby aspirin (Bufferin) every morning with his breakfast, so it doesn't upset his stomach.