I’d say a lot of pet simulator games tend to mirror real life animals the most. Especially, horse games where some of them might pass away if you don’t care for them correctly.
Some pet sites have features in place as well where your pets will tell you that they’re starving or hungry, but generally, there isn’t any consequences involved if you’re not actually taking care of them.
Unless you’re taking care of a digital pet like Tamagotchi or a Digimon, then they’ll pass away on you. These games have natural behaviors like real life animals.