Go to the vet to be painlessly put out of misery?

Does " go to the vet to be painlessly put out of our misery" mean "to receive a mercy killing/euthanasia"?

Context:

Why don't religious people talk like that when in the presence of
the dying? Could it be that they don't really believe all that stuff
they pretend to believe? Or perhaps they do believe it but fear the
process of dying. With good reason, given that our species is
the only one not allowed to go to the vet to be painlessly put out of
our misery. But in that case, why does the most vocal opposition to
euthanasia and assisted suicide come from the religious? On the
'Abbot of Ampleforth' or 'Holiday in the Seychelles' model of
death, wouldn't you expect that religious people would be the least
likely to cling unbecomingly to earthly life? Yet it is a striking fact
that, if you meet somebody who is passionately opposed to mercy
killing, or passionately against assisted suicide, you can bet a good
sum that they will turn out to be religious. The official reason may
be that all killing is a sin. But why deem it to be a sin if you
sincerely believe you are accelerating a journey to heaven?