In Madagascar, Alex’s birth name is Alakay.
The name is revealed in Madagascar: Escape 2 Africa, when Alex remembers his childhood in Africa. As a young lion, Alakay is separated from his father, Zuba, after being captured and transported away from the savanna. He eventually reaches New York, where he becomes the zoo’s famous star under the name Alex.
This detail is easy to miss because the first Madagascar film uses Alex almost exclusively. The sequel connects the character’s New York identity with his African family history, including his mother, Florrie, and his father’s rivalry with Makunga.
Alakay is therefore not a second adult identity or a nickname used by the other zoo animals. It is the lion’s original name before his life in captivity and before he becomes known worldwide as Alex.