Unlike doctors, vets euthanize animals every day. They're comfortable with the idea of a medically induced death. Ask any vet if euthanizing animals makes them uncomfortable and they'll tell you it doesn't bother them at all because they know they are ending the animal's suffering humanely.
On top of that, they have access to lethal drugs and know precisely how to use them, how to end their life quickly and painlessly.
Those two together make it very easy for me to see how a vet would rationalize suicide.
And I bet you rarely encounter awesome, happy pets. You might get the occasional cute puppy/kitty in for theirs shots (which the vet techs probably do), but I bet most of the pets you deal with are sick and/or dying.
Even if you grow callous to the death of animals, encountering the related human suffering must be miserable. I imagine the people most likely to take their pet to a vet are the ones most attached to them, therefore the most affected by their suffering and death.
On top of that, they have access to lethal drugs and know precisely how to use them, how to end their life quickly and painlessly.
Those two together make it very easy for me to see how a vet would rationalize suicide.