Kovač was born in Šibenik during the Italian occupation of Dalmatia in the World War II as the second child of Zrinka and Jakov Kovač. His mother was originally from the island of Vrgada, and his father was from the Škopinac district of Šibenik. Mišo grew up in a family with his brother Ratko and sister Blanka. As a child, he grew up in a street in Šibenik's Varoš, where musicians Vice Vukov and Arsen Dedić lived at the same time. As a young man, he practiced football as a goalkeeper and started playing for the junior team of HNK Šibenik, dreaming of a professional football career in Hajduk Split, which he was also a fan of, often traveling by boat from Šibenik to Split for matches. He was receiving defeats of his team very emotionally.