Interesting post because it's one of several I've seen lately about situations where the poster's home is the hang-out spot for neighborhood kids -- and the hanging out goes on for hours and hours, even on school days. Not specifically arranged play dates for specific kids, just kids who arrive unannounced and stay. I'm not sure when any of these kids are getting homework done--?
There's a bigger issue here than the question about his being a guest. He's not a guest. He's a resident.
He may be a nice boy, but he's not your son, so I wonder what is so lacking at his own home that he is camping at yours? That you are feedling him full meals weekly? Do you also feel free to discipline him if he misbehaves? Do you have authority to make him stop doing something, or to order him home? (I think you do, but his mother apparently would think otherwise.) And she surely would be quick to come to you if her son were hurt at your house -- you would be legally liable.
I would end the after-school babysitting and the all-day marathon weekend babysitting (that's what it is, unpaid babysitting with you providing the meals) and the next time he turned up early on some Saturday I'd smile at him and say, sorry, we're on our way out the door, and we'll drop you back at your house on the way. Then DO that. I get that you like being part of the "village" but that idea includes mutual respect for the other villagers; his mom shows zero respect for you when she simply assumes that you will be available for her child. After all, the moms of other local kids don't do this, do they?
I think his mom is blatantly using you for huge amounts of free babysitting, frankly. She is using you, though she may not even realize it, and he is learning to use you through her example and your family's constant availability.
Time to break what has become a habit for a very manipulative family and tell them your children need after school for homework time and their own activities; and be going out the door the next weekend day he just shows up. Tell him he is very welcome to come over but needs to ask first. It may sound cruel but actually it would be a kindness to him; he needs to get to know his own family and they need to take responsibility for him. Dropping in whenever for however long you'll let him? That's not being part of a village, that's being rude. You do not have to let this child practically live with you in order to be hospitable to him.