When Buono da Malamocco and Andrea "Rustico" da Torcello left Alexandria for Venice with the purloined remains of Saint Mark in 827, they brought with them all the iconographic paraphernalia associated with the evangelist, including, of course the lion, that, in turn, came to represent the city.
There is a whole language associated with the winged lion of Venice--paws on land or on water, wings spread or wrapped around its head ("leone in moleca",) book open or closed, sword drawn or the ground, with or without halo--representing different aspects of St. Mark's story, as well as a variety of circumstances connected to the Venice Republic, "La Serenissima", its forms of government and its territory.
Interestingly enough, neither the winged lion nor St. Mark were the first emblematic figures of Venice. The canonical view of the Venetian lagoon from St. Mark's Piazzetta is framed by two columns. The one of the left has the famous winged lion on top, but the other one is crowned by a figure of St. Theodore, the patron saint of Venice before arrival of St. Mark's remains. And with St. Theodore comes his dragon, here represented by a beast looking more like a crocodile.
(Talk about branding and rebranding!)