In the story of Joseph and his brothers, we are told not to play favorites. How then could G-d play favorites by choosing the Jews?

As you correctly point out, the relationship between G-d and the Jews could not be one of favoritism. Israel's special relationship is due to a reality - a fundamental difference between its identity and that of Gentile nations - which distinguishes it from the other nations. This said, it is possible for someone to join in this special relationship by accepting the Torah and thus, by converting to Judaism, joining the Jewish People.