No, they were not Israelite's and that is the point.
He answered, “It isn’t right to take the children’s bread and throw it to their dogs.”
Now, that statement wasn't for her. It was actually for His Jewish audience. At that statement, His Jewish audience undoubtedly nodded their heads in agreement, because they were bigots against gentiles, particularly Canaanites and even Samaritans (who were not, technically, gentiles). His disciples had already tried to send her away, even though they knew Jesus was in the exorcism business.
But then Jesus did exactly the opposite thing (and this was not the only time Jesus appeared to be leading His audience down the path they expected, then made an unexpected turn).
Then Jesus replied to her, “Woman, your faith is great. Let it be done for you as you want.”
Did Jesus change His mind? No, He didn't.
He demonstrated a principle that Paul explains in detail in Romans 8 and 9: It was never lineage, it was always faith that made anyone a child of Abraham.
From Jesus' point of view, it was always, "If you've come to me in faith, you are a child of Abraham, and I am for you."