Florentine was first used as a term for a weapon style within the Society for Creative Anachronism circa A.S.2 (1970 AD) to describe a fighting style involving the use of two pounds of spinach and a pair of salad forks. Later the spinach was either discarded or eaten (feasts often started late in those days) and the term came to denote any two-weapon style, or, alternatively "what medieval knights would have called fighting in tournaments with two weapons at once if they had ever done such a thing, which they didn't". The style is sometimes referred to as “Too many swords.”
While medieval men at arms sometimes carried a second sword in case their primary weapon was lost or broken, there is no evidence they fought in armored combat with two at one time.
There are three main sources of inspiration for the use of two swords or two weapons in the Society’s recreation of medieval armored combat. The Icelandic sagas sometimes describe characters fighting with two weapons. In Njal's Saga, for example one character is bushwacked while cutting firewood, and fights with his axe and his sword. The hero Gunnar often fights with his sword and his magic "halberd". (Since the story is supposed to be happening long before what we think of as halberds appear, it's a little unclear what the weapon actually was). One could argue that these were special cases, and that the Sagas are not entirely dependable as factual evidence. But even taken at face value, they are not evidence for the use of "two weapons" in tournaments, since the action occurs before the first tournaments are invented.
The second is renaissance styles of unarmored combat that used either sword and dagger or, more rarely two swords.
The third is a fighting style using two swords introduced by the Japanese swordsman Miyamoto Musashi some time after 1600.
The early 15th c. fighting manual Flos Duellatorum by Fiore dei Liberi does include a brief sequence on fighting with due bastone or two clubs. This seems to be an example of how to improvise a defense in unarmored combat, like a similar sequence on how to defend yourself using only a walking stick and a dagger. The clubs are rude and unshaped tree branches, the attacker is armed more conventionally, and the sequence ends with the defender throwing away one of the clubs.
Fighting with two swords at once can be a reasonably effective technique using Society armored combat rules, but the combat style was almost or entirely unknown in medieval armored combat. The difference probably stems from a mismatch between the combat rules and reality. In SCA armored combat, swords are virtually unbreakable and hands are considered to be invulnerable to attack. Neither was true in the actual Middle Ages.