Chad Gadya is a Pesach song because it speaks to the isolation of the Jewish people, symbolized by the kid. The kid is both a representation because of its helplessness and the Israelite history as shepherds.
In Exodus, the commandment is to serve a kid to each family group and to daub the blood on the doorposts of the Israelites.
Finally, in the Commandments handed down at Sinai, the instruction is that thou shalt not seethe a kid in its mother's milk.