Not to bust in as some sort of grammatical know it all, or something... but my assumption of why exactly Petrie spoke the way he does was as a play on his nervous nature. Using "Me" at the beginning of a sentence is one way to portray a character as either less intelligent, unconfident in themselves or less educated in a particular language.
Petrie in most circumstance is shown as quite a smart little fellow, in most cases. The language portrayal is a difficult to pin down, since there is very little discussion over the topic of speaking. Nor is there any case of anyone actually questioning why Petrie speaks the way he does. Sure, he spoke like that when they first met... but that brings me to a more logical explanation.
When we first meet Petrie, he is nervous and unconfident in his ability to fly (as well, supposedly, his bravery). That is portrayed to younger children by his speaking manner, in which we can quickly identify with his nature, showing in the way he refers to himself. It is much easier to identify that a character is nervous and jumpy when they have a manner of speech that matches it. Think, for example, what Petrie would be like if he was the way he was, but spoke in a more literate manner: "I'm Petrie," as opposed to "Me... me Petrie," or the way he reacts when presented with some sort of danger. The words don't exactly come out of his mouth very well in those circumstances.
Short answer: Petrie speaks the way he does because it better portrays his character traits.
