From the Super Mario Wiki
Prince Mushroomkhamen was once the prince of Desert Land, but is now a mummy confined to a sarcophagus. His parents are Queen Mushroomkhamen and King Mushroomkhamen. As it so happens, Mario bears a great resemblance to Prince Mushroomkhamen. As the sarcophagus is never opened, the current, mummified state of Prince Mushroomkhamen is never actually seen.
The prince is stolen by Hip and Hop Koopa, which incites the rage of Queen Mushroomkhamen, who terrorizes Desert Land in a rage. She later sees Mario, mistakes him for her son, and forces him to take his place. In the end, the prince is found and returned to his mother by Princess Toadstool, Luigi, and Toad.
Names in other languages
1 - Mistranslation in one Hip Koopa's lines at the beginning of the episode. Prince Mushroomkhamen is subsequently referred to as "prinț" ("prince") throughout the episode, though never by his full name again.