So have a headcannon.

To me, when a cat in Warrior Cats is reincarnated, they first go to StarClan. There they will meet with important cats in their life as normal. If reincarnation is possible, one of the important cats will bring them as a ghost to the real world and present the possible kits. There, the cat will chose a kit. The important cat will press their nose to the reincarnated cat's and the cat's spirit will forever be able to follow the kit.

As the kit grows and progresses through life, the spirit will be with it every step of the way. They will feel the cat's emotions, and know everything that happens.

When the spirit wishes, they may reveal themselves to the cat in a dream. There, the spirit will introduce themself and thank the cat for allowing it all to happen. The spirit will then return to StarClan.

Cats are usually reincarnated if they died early on or just didn't have a fulfilling life. Through the chosen cat, they will be able to "re-live", and get to live like they wanted. Of course, the spirit cannot choose what the cat does, but they can observe and basically live alongside them.

A cat can only be reincarnated once.