For domesticated cats and dogs, being inside the home
is their natural environment (or can be, to varying degrees).
I've owned cats and dogs. I can let them outside and leave the door open, and they may go outside to explore or play briefly, but they'll want to come inside shortly afterward. Especially my dog - he doesn't even want to be outside by himself, except to relieve himself. He has herding instincts and is uncomfortable being alone outdoors for more than a short time. He would much rather be around people, but if he had to be by himself, then he'd rather be indoors. (Observation: if I were to leave the house, and left the back door open, he'd still spend most of the day inside rather than in the yard.) My cat acts downright frightened being outside.
It's not really possible to discuss what is "natural" here, and I'd recommend being cautious about the naturalistic fallacy [1]: the idea that because something is 'natural' it must be better. It is 'natural' for humans to die of all kinds of diseases, yet we do not wish to, and so we treat ourselves with antibiotics and other medicines. It is 'natural' for children and women to die in childbirth, but we take substantial steps to prevent that, and so on.
In the case of domesticated animals, it's hard to reason about what "natural" even is. The animals have adapted to living with humans; they want to be around humans. They aren't feral, and wouldn't want to live by themselves in the wilderness any more than a human would. They have a choice every time I open the door.
(And yes, if they ran away, I would feel obligated to find them, but that's because I'm responsible for their health and safety, not because they are my prisoners. The point is that they don't even come close to making the choice to do that. It is far more likely that they would simply get lost outside than actually intentionally run away. This has happened once before to my dog, and his emotional reaction upon me finding him demonstrated compellingly that he felt lost and was happy to be found and reunited.)
I might feel conflicted if I ever had a pet that behaved like it wanted to escape from me, but that's never been the case. Even pets that do enjoy to spend time outdoors alone - and I'm happy for them to do that - always want to return to the safety of the home after a time. I would not enjoy having a pet if I thought at all that it felt like a prisoner.
Lastly, I would remark on the fact that these animals were raised from infancy by humans and around humans. They are socialized to be part of the family, just like people are. While it is true that they would have different personalities if they were raised in the wilderness, the same would also be true of a human child. If a human child were abandoned alone in the wilderness and somehow survived, then this human would undoubtedly have great difficulty socializing as an adult, and fitting into a family - yet we would not say that this wild person is somehow better or more natural. We would say that their upbringing was cruel.
[1] https://en.wikipedia.org/wiki/Naturalistic_fallacy