Examples of how medications vary between dogs, cats, humans.
A good example is levothyroxine. Levothyroxine in humans is dosed in micrograms, in dogs it is dosed in milligrams. Large dogs can easily take up to 0.8-1.0mg of levothyroxine twice a day. Dogs can tolerate really large doses of levothyroxine and need higher doses for treatment and management of hypothyroidism.
Another example is acetaminophen. Humans can tolerate some fairly decent doses compared to dogs and definitely compared to cats. Dogs can tolerate small doses of acetaminophen, but it is generally not a recommended pain control for dogs. Cats can't tolerate any amounts of acetaminophen. Acetaminophen is metabolized in multiple different ways, one of which is glucuronidation. Glucuronidation creates a non-toxic metabolite, cats have significant lack of the enzyme that does glucuronidation. While cats can metabolize the acetaminophen via the sulfation pathway (to non-toxic metabolites), they also have a reduced amount of this pathway and once this pathway is saturated, the acetaminophen is then broken down into the toxic NAPQI metabolite by cytochrome p-450. Cat's red blood cells are really susceptible to the effects of NAPQI and cats very easily develop methemoglobinemia when given acetaminophen.
NSAIDs are another great example. Human medicine has great NSAIDs: ibuprofen, naproxen, diclofenac, celecoxib, indomethacin, etc. Dogs and cats generally can't tolerate human NSAIDs and they can develop GI ulceration and acute kidney injury. We do have some veterinary specific NSAIDs for dogs and a couple of ones for cats. Cats in general are very sensitive to the effects of NSAIDs, even the veterinary specific ones we monitor blood work closely in cats and dogs taking NSAIDs chronically.
Another thing we worry about in dogs especially that isn't worried about in humans are any products containing xylitol. Human medications can sometimes contain xylitol, so we have to be wary of that as well. Xylitol will cause hypoglycemia and potentially liver failure in dogs.
Never hesitate to reach out to a veterinarian if you have a question. Dogs and cats are much different than humans and different from each other. We use some things in humans that should never be given to a dog or cat, and there are some things we use in dogs and cats that aren't and shouldn't be used in humans.