"Euthanize" is a medical term applied to mercy killing, applied to alleviate pain & suffering, hopefully as a better condition for the patient. "Kill" or "killing" is done to the second entity by and for the benefit of the first entity. Intent. This is language.
In animal control situations the dogs and cats etc. are killed. It is termed "euthanized" because it is accomplished by injection or gas in a supposedly clinical setting. It is also a socially acceptable term. It may alleviate some starvation, injury or discomfort to animals on the streets. But the major beneficiaries are the human residents of the community that do not have to deal with the inconvenience of cleaning up poop or entertaining large groups of suitors when their animals are in heat. The animals in the shelter would prefer to live.
Hunters generally do not euthanize their prey , they kill them for food, trophy or the experience, not necessarily in that order. Taking/killing a very old, limping, scarred & solitary bull or buck yields a storied trophy. Despite having danger or requiring stealth, this is when killing may cross the line to euthanasia. 1 shot. 6 shots is another whole thing....