Long long ago, pubs didn't have names but they just had signs. People would call the pub whatever was on the sign. "The King's Head" for pubs with a portrait of a king, "The Wheat Sheaf" for ones with a picture of some wheat or barley, etc.

Lots of old pubs displayed the Stuart coat of arms as a show of loyalty to King James I/VI and his heirs, which is a heraldic red lion. Hence why so many pubs have the same name even though they're all ancient and unrelated.