I've lived in Florida for decades and I have a hard time answering that question. As close as I can come is to point out that it is the Florida Department of Agriculture and Consumer Services. Surveyors, fair rides, auto repair, and private investigators come under their purview. I'm not saying that concealed weapons permits necessarily fit, only that they have a portfolio of oddball stuff. It seems like the "consumer services" part makes the legislature dump stuff there that doesn't fit somewhere else. Not a great explanation, but the best I can come up with.
I will say, the relatively recent capability of renewing CWPs at local tax collector's office makes it fast and easy. My local tax collector has a separate little private room they take you, and you walk out with the laminated photo ID in your wallet.