Hello, I have had my cats for 13 years and would be most upset and angry if one of my neighbours gave one of them to a rescue centre if they mistakenly took them for being lost. I would think you have every right to reclaim your cat, but I am wondering what attempts you made to find your cat, when it went missing? In your neigbours defence, perhaps she expected you to knock on doors to ask people if they had seen your cat? Perhaps this is why she assumed that the cat was a stray?
When one of mine went missing, I wrote notices and put them through all my neighbours doors, so they would be in no doubt who owned it. Thankfully she was back within 24 hours. I am not trying to sound judgemental, but just trying to give another opinion and perhaps a reason for why she took the steps she did. Best of luck though, Sue.