No—the Sega Mega Drive (renamed Genesis in North America) was not named after an animal. The "Mega" prefix came from Mega Man (Rockman in Japan), Capcom's hit franchise that debuted in 1987, two years before the console's 1988 Japanese launch.
The Mega Drive/Genesis name is a marketing fusion of a hit game character and tech-forward terminology—not a lion, fox, or hedgehog.