We know that we don’t truly own our domestic cat companions. We might think that we possess them but we don’t even do that if they are allowed outside. This is the correct state of affairs as cats are sentient beings and the best description of our relationship is not one of ownership but of mutual benefit. There are countless examples of cats choosing to live with a neighbour rather than remain in their home with their ‘owner’. There are many possible reasons and I’ll mention three.

Food
This is the obvious and strongest reason why your cat might leave you. Your neighbour provides better quality food. Your neighbour might not provide food deliberately to your cat but your cat goes into a neighbour’s home. Your neighbour has a cat who she feeds very well and she allows your cat to share that food. She does this because she likes cats and can’t say no. And she likes the visit. It brightens up her life and adds a bit of variety. And even her cat likes your cat as well so it’s ‘a result’ as they say in the football business.
Presence
Cats need us as companions. That is, they need us to be around. They need our presence. There are some people who still think that cats are independent and don’t mind whether we’re there or not but they are wrong. Your neighbour is retired while you are working all day every day sometimes for long hours. Your cat finds company with your neighbour and they strike up a friendship which fills your cat’s day.
Cat companion
A less likely additional reason is that strikes up a friendship with your neighbour’s cat. It’s possible that your neighbour’s cat comes to your home then when they go home your cat follows. There is a merging of homes which might lead to a gradual migration of your cat to your neighbour’s home.
This scenario indicates that domestic cats can be quite sociable, which they are nowadays. It depends upon the individual cat of course. The right cat companion for a domestic cat is better than a human companion in terms of pure companionship. Obviously a cat companion can’t provide food and shelter but in terms of pure friendship it can be superior to the cat-to-human friendship.
Two neighbours
A cat might even choose two other neighbours to share their lives with. There is an interesting story on the Quora.com website written by Andrew. The neighbour’s cat used to come into their home during the daytime when they weren’t there to eat their cat’s food. The neighbour’s cat was called Barát by his owner. They called him Doofus. They liked him. After two years Doofus “began hanging out with the old man who lived down the street. He was home during the day and they kept each other company. Doofus rarely returned after that. The old man named Doofus to Walter and the two passed away in the same month a few years later”. This was a form of consecutive time share. Cats can time share concurrently as well sometime without the ‘owner’ being aware of it.
Andrew says that Barát/Doofus/Walter, “chose the right home for himself at different times in his life; a freedom few pets enjoy”.
Conclusion
As I said, the concept of human ownership of an object does not work well in the cat-to-human relationship. This is because cats have a will of their own. They can choose if allowed to.