"And in his needy shop a tortoise hung,
An alligator stuffed and other skins
Of ill-shaped fishes..."
--from Romeo and Juliet, by William Shakespeare, describing an apothecary's shop.
"There the wizard sat in all his state. A stuffed crocodile canopied his head, a serpent's skin of large dimensions was spread under his feet..."
--from Anastasius (1836), by Thomas Hope.
The Century Magazine in 1914 lists the stage properties of a wizard as "his gown and wand, the stuffed crocodile, and the skeleton in the corner."
"There were tables strewn with archaic instruments of doubtful use, with astrological charts, with skulls and alembics and crystals, with censers such as are used in the Catholic Church, and volumes bound in worm-eaten leather with verdigris-mottled clasps. In one corner stood the skeleton of a large ape; in another, a human skeleton; and overhead a stuffed crocodile was suspended."
--from "The Return of the Sorcerer" (1931), by Clark Ashton Smith.
"There was a real Corkindrill hanging from the rafters, very life-like and horrible with glass eyes and scaly tale stretched out behind it. When its master came into the room it winked one eye in salutation, although it was stuffed."
--from The Sword in the Stone (1938), by T. H. White.
"Gradually, the children discovered other treasures; a chart on which the signs of the zodiac were nicely touched up by Miss Price in water colour; a sheep's skull; a chocolate box full of dried mice; herbs in bunches; a pot of growing hemlock and one of witch's bane; a small stuffed alligator, which hung by two wires from the ceiling.
'What are alligators used for, Miss Price?' asked Paul.
Again Miss Price's training in truthfulness overcame her longing to impress. 'Nothing much,' she said. 'They're out of date now. I like to have it there for the look of it.' "
--from Bedknob and Broomstick (1943, 1957), by Mary Norton.
"The marquis took in the room, eyes sliding from detail to detail. The stuffed crocodile hanging from the ceiling, the leather-bound books, an astrolabe, convex and concave mirrors, odd scientific instruments; there were maps on the walls, of lands and cities de Carabas had never heard of; a desk, covered in hand-written correspondence."
--from Neverwhere (1996), by Niel Gaiman.
"Like all wizards' workshops, the place looked as though a taxidermist had dropped his stock in a foundry and then had a fight with a maddened glassblower, braining a passing crocodile in the process (it hung from the rafters, and smelt strongly of camphor)."
--from The Light Fantastic (1986), by Terry Pratchett.
"There was a large crystal ball with a crack in it, an astrolabe with several bits missing, a rather scuffed octogram on the floor, and a stuffed alligator hanging from the ceiling. A stuffed alligator is absolutely standard equipment in any properly-run magical establishment. This one looked like it hadn't enjoyed it much."
--from Mort (1987), by Terry Pratchett.
"Because a raven sitting on a skull and going 'caw' is as much part of your actual wizarding modus operandi as the big dribbling candles and and the old stuffed alligator hanging from the ceiling. Don't you know anything? I should have thought anyone knows that who knows anything about anything. Why, a proper wizard might as well not even have bubbling green stuff in bottles as be without his raven sitting on a skull and going 'caw'--"
--from Soul Music (1995), by Terry Pratchett.
"It was a wizard's study, so of course it had the skull with a candle in it and a stuffed crocodile hanging from the ceiling. No one, least of all wizards, know why this is, but you have to have them."
--from Going Postal (2004), by Terry Pratchett.
So, why do you have to have stuffed crocodiles?
There are two answers, the long one and the short one.
The short one is that they are cool and unusual, rather wicked looking, and give the impression that something out of the ordinary is going on where it is. This is why they are in bars and souvenir shops to this day.
The long answer is historical, convoluted, and a little speculative, and it goes back to the Middle Ages.
In those days, the main type of private travel was the religious pilgrimage, where citizens of Christian European countries would travel to the Holy Land in the Middle East to accrue spiritual merit. Like folks travelling in any age, they would bring back mementos of their journey, the more peculiar and exotic the better. Stuffed crocodiles were very popular, being unfamiliar in the cold northern climes; private citizens would bring back smaller specimens, whereas kings and other potentates could receive or import quite large examples. The impressive size of these could give rise to tales of dragons as their true origins receded into the past. Sometimes living examples would be taken home, and if they escaped in the warmer countries (like Spain or Italy) might have survived in a wild state, gone rogue, and have to be hunted down. There was one king who kept a crocodile in his moat, which may have given rise to the whole idea of moat monsters.
In any case, stuffed crocodiles became associated with the studies of theologians, natural philosophers, alchemists, apothecaries, and other studious sorts, in the days when these disciplines where not greatly distinguished from each other. Even surgeon-barbers might have a specimen hanging around to advertise their expertise: the beast's toothy grin could well have a customer rubbing his own jaw. Crocodiles would be figured prominently in Cabinets of Curiosities, chests or even whole rooms filled with peculiar objects and interesting specimens. With the discovery of the New World, alligators joined the crocodile among the curios.
But with the progress of travel, trade, and discovery, the big lizards became less and less unusual, fell out of fashion, and became relegated to second-hand shops and less progressive establishments. The memories of their association with more mystical, less scientific times persisted, however, and their use as props in magic shows kept this memory alive long enough to become part of the wizard's tradition in popular literature.
Pictures: 1) A Cabinet of Curiosities; 2) The Alchemist At Work by David Teniers; 3) John Dee and Edward Kelly; 4) Merlin's House, by Alan Lee, from The Sword in the Stone; 5) and 6) from Bedknob and Broomstick; 7) Sokurah's Workshop from The 7th Voyage of Sinbad, and 8) A Discworld Wizard by Dan Pence.