So, there seem to be two separate questions here: “should enemies always fight to the death?” and “what should enemies do when a PC is unconscious?” I think the answer to both depends on the situation, and the enemy.
A wild animal (or a monster with animal-like behavior) will usually not fight to the death unless it has no other choice. In my games, a single damaging hit is enough to scare most Beasts into retreating, unless they’re rabid or defending their young. Also, most such creatures usually wont continue attacking an unconscious PC. Maybe if its goal is to get food it might try to drag an unconscious PC off, but most of the time if you’re fighting an animal it’s because it’s defending its territory, in which case killing unconscious PCs doesn’t really move them closer to their goal.
Humanoids and monsters with humanoid behavior usually don’t fight to the death either. Usually if they think they are loosing, they will attempt to retreat if possible or surrender otherwise. If their surrender is not accepted, then they will fight to defend themselves, but the goal is “escape with my life,” not “kill my enemies,” so that’s going to affect their tactics. Humanoids might kill unconscious PCs, try to take them hostage, or ignore them, depending on their goals. Assassins sent to kill the PCs risk a lot for little gain by trying to hold unconscious PCs hostage, but they gain a lot for little risk by killing unconscious PCs. On the other hand, bandits trying to rob the PCs most likely want their money, not their lives, so holding an unconscious PC hostage is a more effective strategy than killing them. An opponent who is fighting merely to survive against PCs who want to kill them might attempt to take a hostage if they have reason to think the conscious PCs will respond well to that and might kill an unconscious PC if they have reason to think an unconscious PC is still a threat. But most often the priority in that situation is just to escape; an unconscious PC doesn’t pose an obstacle to that goal, and a dead PC motivates their surviving companions to seek revenge.
Mindless creatures usually follow directives given by some kind of master or other motivating force, and follow those directives literally, with no room for interpretation. These are the opponents most likely to fight to the death because they have no sense of self-preservation. They relentlessly carry out their instructions until the task is completed, or until they become physically unable to carry them out. Whether or not they kill unconscious PCs depends on their instructions. “Kill anyone who enters the sanctum” will definitely lead to them killing unconscious intruders. “Bring those meddling fools to me alive” will definitely not.
For opponents with more alien behavior patterns it can be a little trickier, but in general you can work out their most likely course of action by considering their goals and what they are willing to risk in pursuit of them. Combat is a means of resolving conflict, and conflict is a result of two mutually exclusive goals coming into contact, so in a combat, it’s always important to assess what the PCs goals are, what the monsters’ or NPCs’ goals are, and how far each party is willing to go in pursuit of them. This should give you a pretty good idea of whether or not the creatures will fight to the death and what (If anything) they will do about unconscious PCs.