"Wildcat" is a term used by amateur ammunition reloaders to signify some sort of either experimental nature or enhancement that is the result of any number of ammunition manufacturing processes. Typically, wildcatting is only done with brass cartridges and this means it is done with pistol and rifle ammunition and not shotshells. That said, typically a wildcat cartridge is for a rifle and not a pistol. This is not to say that wildcat pistol rounds do not exist as they certainly do. One such example is the .38 Super, which was based on the .38 ACP.
Wildcatting of cartridges is done for any number of reasons. Some people, perhaps defective in character, like to have their name attached to a specific cartridge that is "faster, whiter and brighter"; some people wildcat a cartridge because they believe they can make a better ballistic device and some people like to tinker for its own sake. Whatever the reason, just such a cartridge and person are both called a wildcat or wildcatter, respectively.
At any rate, the reason shotgun shells are not typically developed into wildcats is because shotguns are not usually able to withstand pressures that a rifle or pistol can typically handle; they aren't subject to much room in experimentation as a result. This is not directly due to the materials used; rather, it is both because of the design of the modern day shotgun and the steel used to construct said. Since shotguns are not designed to withstand high chamber pressures, they can explode if the shell goes beyond the pressure (measured in PSI) that it was made to accept (say, 11,500 PSI), or the level of pressure to which the gun is "proofed." The modern rifle and pistol are manufactured to higher pressure tolerances because their ammunition requires higher working pressures (say, 30-50,000 PSI) and their internal designs and materials are selected accordingly.
Wildcat rounds are manufactured from brass that is already commercially available, however archaic the original cartridge may or may not be. Some of the processes used to create a new wildcat include, but are not limited to, flaring out the neck, reducing the angle on the shoulder, increasing the angle of the shoulder, trimming the cartridge to a new size and performing any of the above steps together. Then, there is the ballistic testing and questions to be answered — how accurate, how fast, what is happening to the brass during the firing sequence, are the primers popping out showing too much pressure, do the cases separate? Notes are taken. The whole process can take as long as years and as short as months. It just depends on the steps required to prepare the brass in addition to the field testing that must be performed to gauge success. In the end, a wildcat must change the cartridge in such a way as to give it unique properties unto itself.
Occasionally, the big ammunition manufacturers (Federal, Remington and Winchester) will take note of an excellent wildcat and begin to commercially produce it. One such case is the .22-250 Remington. When this occurs, the round is no longer considered a wildcat.