Moshe, the Hebrew
How well did Moshe know his Hebrew origin?
The text clearly shows that Moshe knew he was a Hebrew; let's give some examples:
His foster mother recognized him as such immediately. This must mean there was something about him that made him look Hebrew. What could it be, in such a small baby? The color of his skin, maybe—it would be blatantly different from the Egyptians in general?
When he kills the foreman, his first reaction is not "oh, grandpa will understand; after all, he was just a foreman and I, his grandson, son of his daughter, I am the prince of Egypt! Everything will be alright!". On the contrary, he flees because he was certain this killing was not something the Pharaoh could tolerate, lest all remaining Hebrew would take that as an example and provoke an insurrection.
When God tells him, in the Burning Bush episode, what He wants Moshe to do, He doesn't have to explain "because you are Hebrew yourself, too, Moshe"—even though He is made to explain lots of other things by Moshe ("lazy to speak" cannot be the right rendering of the Hebrew ...: Moshe spoke—and a lot!).
Then later, when God tells him to meet Aharon, his brother, no introduction is described: instead, they simply kiss as intimate brothers.
Therefore, a question remains: how did Pharaoh allow a Hebrew child to be raised in his own palace?
Maybe he was not personally, if it is possible to use his word in this context, invested on destroying the Hebrews, that is, he didn't hate them, for example. That was solely a strategic governamental decision.