The neighbor's cat wants to live with me

A cat showed up on my balcony two weeks ago. I see him through the window, so I go out to pet him. He’s incredibly affectionate. The door is ajar, so he wanders into my apartment to explore a bit before heading back out. A few days later, he’s back. It’s the same scenario, except this time he makes himself more at home. I never feed him, yet he still hangs around. Today, he came to visit at dinnertime and then again in the evening. He’s been meowing loudly outside my door for 20–30 minutes straight. I let him in, and he’s acting exactly like he’s my cat.

He isn't wearing a collar, but I think he belongs to the neighbor across the way—someone I don't actually know. It’s 10 p.m. now, and he’s still meowing outside my door. He doesn't seem to want to be outside at all. I don't know what to do...