“It is not right to take the children’s bread and toss it to their dogs.” (Matthew 15:26)

Why is Jesus saying this?

Here Jesus makes a statement to both his disciples and the Canaanite woman who had came to Jesus and said:
"Lord, Son of David, have mercy on me! My daughter is suffering terribly from demon-possession," (Matt. 15:22)
To which Jesus’ disciples urged Jesus to “send her away” (Matt. 15:23).

And Jesus remarked that:
"I was sent only to the lost sheep of Israel.” (Matt. 15:24)
Then the woman knelt before him, and said:
“Lord, help me!” (Matt. 15:25)
So Jesus is responding to his disciples’ remarks about sending the woman away. Why is he talking about taking children’s bread and tossing it to the dogs then?

This is an analogy. The "bread" is referring to his teachings.

And the "children" are those who are to receive his teachings.

While the "dogs" are those who are not ready or unwilling to receive his teachings. (During those times, people did not feed dogs their dinners as they do today.)

A teaching of compassion

Jesus is chastising the lack of compassion exhibited by his disciples. He is attempting to give them a lesson about his mission and his objectives.

To Jesus’ statement above about the dogs, the humble woman says:
"Yes, Lord, but even the dogs eat the crumbs that fall from their masters' table." (Matt 15:27)
In other words, she submits herself in full humility and faith to the representative of God, not caring to feel offended by being compared to a dog (which Jesus was only doing to illustrate how his disciples were treating the woman).

To this we will find that Jesus praises her humility and faith, and heals her daughter without any hesitation (see Matt. 15:28).

As the loving servant and representative of God, Jesus is compassionate. He cares about everyone - not just the Jews.

Jesus did not identify the person with the physical body. He knew our identity as the spirit-person within these temporary physical bodies. This is why he also said at one point:
“Do not be afraid of those who kill the body but cannot kill the soul. Rather, be afraid of one who can destroy both soul and body in hell." (Matthew 10:28)

The physical body is temporary

The physical body will live for a few decades at the most and then die. It is the spirit-person within that Jesus was considering, and who he was teaching this to.

Thus the position or family of one's body was not his concern. Rather, the condition of the person's heart. Do they have faith in the Supreme Being? Are they prepared to listen to the teachings of God’s representative and follow them?

Are they prepared to submit themselves to the Supreme Being, even gradually, through their lives, and develop their relationship with the Supreme Being?

These are the issues Jesus was focused upon, rather than the issues of the temporary physical body - the body that will die and decompose in the ground.