I've grown up with cats my entire life and think that I understand them...as much as any human can, like 
We live above a shop. Our flat entrance is at the back, up a fire escape type stair case and our kitchen window is above the annex of the back of the shop. There's a little forest that houses foxes, badgers, other cats and my little cat plays out there every day.
My cat is very distinct looking - she's a Bombay so completely black, massive ears, big gold/yellow eyes, a little miniature panther :) I've seen all the neighbourhood cats and they come round occasionally but never up the stairs. Only the foxes come up to the front door.
My cat sits on the windowsill of the kitchen window to indicate she wants to come in. The other morning I wandered into the kitchen, half asleep, saw a black cat shape, went to open the front door...until I realised my little cat was winding herself in my legs. Very confused, and more than a little freaked out, I realised it was an identical cat sitting on our windowsill. On closer inspection, he's a Tom, big and muscular and he cries to be let in whenever he sees me. He acts like he's lived here, though we've been here four years and never seen him.
Every single night now, he's there - either waiting for us or for my cat. She doesn't seem massively bothered, though we have to chase him away before she'll run in of an evening. He's also there every morning, crying to be let in :(
I feel so bad - my instincts are to feed/cuddle/talk to him. He's a lovely looking thing - he could be my cat's dad, they look so alike. I think he might also be a Bombay, judging from his build. Isn't that odd - to have two Bombay's in one random area?
How/why has he found us/her? She was spayed at six months so although I know Tom's can still hang around, he's not going to get anything. Why has he suddenly started coming around? He's got a beautiful coat and very chunky/well fed so he's domesticated. It's all a bit strange and them looking the spitting image of each other and the way he copies her sitting on the windowsill freaks me out a little 