We have a neighbour that is the same. Whilst I don't have a solution, I can give you my own ideas on the matter and maybe it will help.
I have tried trackers, posters, collars, polite letters, keeping the cat in for months etc. Nothing works. Obviously, I have tried just talking to the neighbour but he always becomes aggressive and shouty, which isn't something I like or know how to rise to effectively... and so I have finally tried the "peaceful" option of just letting it go.
The neighbour did explain to me that "cats just love" him and that this has happened with dozens of cats before my cat - all of whom he has had to eventually rehome for one reason or another. He said he's never had a cat that he hasn't "rescued".
Online, this is something I see a lot of on websites like Reddit or Facebook - that some people just find they "have to rescue" a lot of cats. It's just how it is with some folk - they like cats and they have never had to go out and get one. They perhaps don't necessarily understand that a hungry cat is not a mistreated cat - most cats are hungry all of the time.
There are a few cats in my garden that mill around and whom I could probably "rescue" if I felt like it, but I prefer not to.
My other cat is a neutered male that came from Cats Protection, and who was called Wendy* because his previous owner had "rescued" him from her neighbour. Wendy was still microchipped to the original family and we had to wait 90 days for a request to expire so that we could adopt him formally. We don't call him Wendy anymore - but you see, it happens a lot. Rescue a cat from neglect... rehome the cat after some time. Rescue another.
Please note that I don't think this is the case for genuinely rescued and abandoned cats - it's just that pinching your neighbour's cat is a very common thing!
With this in mind, I will not be reassigning my cat's microchip to my neighbour and I will continue to maintain his vet care / flea treatments etc. when he stops by for a snack. I have a responsibility to my cat, which I will uphold until his very last breath. My neighbour might be feeding him 10 tubs of Dreamies per week, but he isn't responsible enough to look after him properly. And, one day, when he posts him up for rehoming on FB and drives him 50 miles away - hopefully, I'll get a call when they take him to the vet. I have increased the odds of this by setting the cat's microchip to "lost" permanently.
*names may have been changed to protect feline identity.