[Jesus] replied, ‘It is not right to take the children’s bread [the favor being applied to the house of Israel] and toss it to their dogs [the Gentiles].’ ‘Yes, Lord,’ she [the Canaanite woman] said, ‘but even the dogs eat the crumbs that fall from their masters’ table.’”
Although it may at first seem surprising that Jesus would speak with such harshness to a woman in need, his purpose was to draw out her faith. By using the term “dogs” Jesus was mimicking the Jews’ tendency to look down on the Gentiles, for the Jews regularly called the Gentiles “dogs.” Because of their covenant relationship with God, the Jews were ‘typically” clean. All other peoples were not considered clean and styled “Gentile dogs.” Therefore, Jews would never eat, marry, nor have any dealings with them. (See John 4:9.) The Gentiles, in comparison to the Jews, were as dogs of the household. Jesus did not use the word “dog” with indifference or distain, but rather with sympathy, explaining why the woman was not a proper subject for a blessing. She understood her position and with humility replied to Jesus in verse 28, “Yes, Lord, but even the dogs under the table eat the children’s crumbs.” This woman, in her faith, knew that Jesus had the power to heal her daughter. She accepted that Jesus’ primary mission was to minister to the Jews, but knew that there was enough of Jesus’ blessing for her as well. The woman triumphed over the obstacle of the Lord’s argument with wonderful keenness and humility and was rewarded. Jesus told her, “For such a reply, you may go; the demon has left your daughter.” (Mark 7:29)
You asked if Jesus’ comparison of Gentiles to dogs was a correct interpretation. Yes, due to the covenant relationship God had with the house of Israel, it is a correct interpretation. Jesus in his first advent was sent only to the lost sheep of Israel (see Matthew 15:24), that is, those Jews who recognized their sinful condition and were seeking their Messiah. The time for Gentile blessings had not yet come.