Third-Person Limited Omniscient… Maybe
Most of the story is told from a third-person perspective; that is, the story doesn’t seem like it’s told by someone who is involved, but the person still knows what’s going on in Hugo’s head (and no one else’s). It's as if we’re looking over Hugo’s shoulder the whole time, with the occasional peek into his actual thoughts. But we're not looking out through his eyes.
Here's an example from the opening of the story:
From his perch behind the clock, Hugo could see everything. He rubbed his fingers nervously against the small notebook in his pocket and told himself to be patient. (1.1.1)
He's not telling us what he's thinking, but we still know what he's thinking. Whoever our narrator is, he must be a mind reader.
First-Person Central Narrator
But wait a minute. How can this guy be a mind reader?
The only explanation is that this guy is somehow involved in the story, which makes sense because every once in a while, we get a first-person "I" who comments on the story. Of course in the end, we find out just who that "I" is:
Once upon a time, I was a boy named Hugo Cabret… (P2.12.4)
Talk about a magician’s great reveal.