A lion is usually portrayed as a strong animal that symbolizes leadership. Being called Fake Lion means that you're a very fake person. Fake Lions usually likes to show off to others how they are the leaders and the ones with power. They tend to be very fake and would downgrade others to make themselves feel better.
Ever heard of the story of a donkey wearing a lion skin? Google that, and you'll understand.
Alex: I hate staying in that group chat, Joey always likes to act like he's above of others...
Manny: Ikr, he's just a fake lion. But just let him live inside his own bubble.