This is movieverse, just so you know; I haven't yet had the pleasure of reading the book. I own nothing.
The cat could tell you who and what he is. He could do that in a heartbeat, if he really wanted to—he wouldn't even have to speak to you to do it, which is useful considering he can't, not anymore. The only place he can speak is on the other side, and he has no intention of going back there again.
Well, as the cat gets across with a soft paw to the face or a single look, he doesn't need the words of man to speak.
If Coraline was ever to ask, the cat knows he could tell her who he is in a heartbeat. There's no rule against that. Even incapable of the words of man, he could find a way to tell her.
But the cat won't. The cat doesn't want to see the look on her face when he tells—the shock, the disbelief, and ultimately the fear. Fear of him in the face of a child wounds the cat the way nothing else can.
She can't keep me out. No one can. No one even dares to try.
It was a game at first. The cat would like to inform all ignorant questioners that cats who can talk and do things other cats can't should be looked at twice. Such cats should be scrutinized closely. Such cats warn innocent children not to play games with the beldam, but somehow, they never listen.
It was a game at first, and the cat won. It didn't matter. The beldam always lies. Remember that; the beldam always lies.
Now, the cat won't leave, he won't ever leave the beldam alone (not until you give it back to me), and he hates rats more than ever.
The cat has no name. He used to, but doesn't now. This used to bother the cat immensely, but it doesn't matter anymore. He doesn't need a name, not when he moves between worlds with the ease of a dancer taking a leap.
"You think this world is a dream come true, but you're wrong."
The cat protects the children, or, at least, he tries to; someone has to. The cat has lost three times but won when Coraline overthrew the beldam. Now he's stuck; now he is for all intents and purposes silent to man through conventional means. It doesn't matter to him if he's here forever and a half; it's over, it's all over. The children have nothing to fear anymore.
This place is full of secrets. Don't leave me out on technicality. That would be foolish. That's the sort of foolishness that makes you food for the beldam.
The cat could tell you who he is. He could tell you who he is in a heartbeat, and if you're not a child he'd probably take amusement at the flabbergasted look on his face.
But he won't.
You don't need to know that.
There is only one thing you or anyone else needs to know.
The cat protects the children. That's all you need to know.