Who takes the cat?

Long post

Me and my SO moved into a lot house. Theres an actual house in front on a regular single home street and then you go into the back wooden gate and there's a huuuuge back yard and another stand alone house that's ours! You wouldn't notice theres another house there just from looking from the street. So we park our cars in the big drive way and walk behind the gate to our home. We have a neighbor that lives upfront in the first house lets call him Dan. We had neighbors next to us move and left their cat im not sure why but we spoke with them before and they confirmed it was theirs they had for a few years. They said the cat liked to be outside but then it sorta switched and became a fully outside cat. I believe its because they got a dog. Gorgeous cat! And you can tell they took good care of the cat! Always wondered why they left her. After they moved DAN said this is my cat now even though its an outside cat. He put a box on his porch and a cat bowl for food and water and a towel down. He also has a cat inside that he brung home as a kitten-said he found it under his work truck. . . but that cat tries to run out of the house whenever he opens the door. And scratches and rubs at the window when i place my hand on it. Its hilarious..anyways the outside cat sleeps under or on my car or my boyfriends car never under his porch...or box... She or he absolutely ADORES ME! Shes at the gate whenever i come to and from work and she RUNS past my legs and tries to get in my back yard. She sits at my door if we let her back and Will drink water or cat treats i give her. I checked her food Dan sits out a few times and it's always covered in ants. These past 3 weeks the cats front right paw has been up like its injured and im not sure how Dan hasnt noticed. I guess this post is because I want that cat. I want to take it to the vet and get it checked up and get her paw looked at but how do i tell Dan? Im afraid he will say no its my cat...well im afraid of him in general lol I reallllllllly want this cat and we have a special bond i can't explain. I constantly buy her toys/cat scratchers and stuff on amazon. I even named her Luna and hes only called her a cat or been like "have you seen my cat". He even stopped us one day for help because he thought she was stuck on the roof but she was just up there chilling with my boyfriend. He joked that when the major hurricaine last year came he looked for her but couldn't find her because he was gunna let her in the house.The whole time i was so freaking scared and worried and crying but we found her literally the day after it ended and she looked like she found high warm ground to stay safe.

Any thoughts on this situation? Or advice on what i can do? Ofcourse by law I believe it is his cat since he sat out food and water so I want to talk to him first to see how he feels about it.