score:3
I don't think that selling genuine leather follows from stating to be an authorised reseller. Rather I think that at best the sign is saying that the seller has paid the licencing fee to the council of Florence, which legally authorises them to carry out their trade and have a market stall. The idea being that it is forbidden to show up at the market in the morning and pitch your stall anywhere you want unless you are legally permitted to do so.
Although I do not wish to speculate on the genuineness of the permit sign you saw, I wrote at best
because it could very well be that the permit is fake. However neither I nor you are for starters authorised to check this, nor do we have the knowledge and expertise to do so. Nevertheless as a tourist you should indeed care about your seller being an authorised one, since this effectively means that their business is registered and that they are more likely to pay taxes on their earnings. On the other hand an unregistered unauthorised seller is a nobody (and a felon) in the eyes of the law from both a professional and a fiscal point of view. Two problems arise if you happen to do business with the latter: