These bags are very popular in Florence, and I've seen them when travelling in other parts of Italy. There's a store selling them just near the Duomo - I haven't noticed any knock-offs of them in the centre though - perhaps you're seeing knock-offs overseas?
Bloomies used to carry it years ago but no longer. I believe there are a few websites on-line with authentic ones (like maybe 1 or 2). They are really beautiful bags. I had one stolen in Florida once and never replaced it.