As the son of two vets, I have to object to this meme
Euthanasia (of animals, at least) is almost always only done when letting the animal stay alive would only prolong their suffering. It is quick, painless, and very humane. It is a very sad experience for all involved still, but it would be animal cruelty to allow their suffering to continue. Some people abuse the system (some people brought their dog to my parents' to be euthanized for a survivable leg injury), but I think most vets have the sense to know when this happens and stop it (the dog with the leg injury I mentioned was not euthanized and became one of our family pets).
As for human euthanasia, is that even a thing?