NY is a license type card identifying the holder, with another card listing those handguns in possession. A NRA approved training course is required for a basic premisces only or target/hunting license, with a more intensive course required after one year for a unrestricted ccw.
Florida, for out of state reciprocal carry purposes only, and then only with some other states, requires you to hold a ccw in your home state while in-state Fl ccw permits have more stringent requirements, I believe. It is also a license type card.
I believe that dual ny/Fl ccw permits allow for reciprocity in about 41 of the continental states (may be more or less..). Some states do not recognize ccw permits from other states because the other states do not recognize theirs. A example, I believe is Georgia and ny: ny does not recognize the Ga. permitting process as being as stringent as the ny process, so they don't accept Ga. permits for reciprocal carry. In turn, Ga. tells ny to stick it and doesn't recognize the ny permit. Had a Sheriff Deputy friend of mine get into a uncomfortable situation in Ga. when questioned by a Ga. Conservation Officer regarding the handgun he was using to hunt with (Contender) as it required licensing in Ga. (at least for out of state hunters) but then almost got badly burned for carrying a ccw without reciprocal license.
There is a publication available, and updated regularly I beleive, which details the handgun permit laws in the different states and also provides (I believe) detail about reciprocity in different states.
Of course, it goes without saying there are many places you cannot go with a handgun or rifle, even a ccw, like post offices, theaters, some restaurants, schools, macdonalds, some churches, some motel/hotels, as these places prefer unarmed victims in attendance only, so be aware that if you have to shoot your way out of a voting booth, burger king, popeye's chicken emporium or a motel 6 you may be in for more trouble than dysentary.......