However up until this point, it was considered all the same art with different flavors based on individual teachers. One can argue that's all a style is anyway, but is that enough to formalize it as an independent martial art? Unlike Japanese martial arts, which have been divided into formalized styles as early as the 15th century, Okinawan martial artists all traveled and cross-trained to pursue their personal growth as practitioners. Even today, there is a spirit of unity among karate practitioners, culminating in the annual Karate Day celebration (commemorating the 1936 meeting when prominent Okinawan masters came together and officially adopted the term Karate 空手 to describe their art, along with the adoption of the kyu/dan rank system, shogo titles and competition format to conform it into a Japanese martial art).
This brings up a rather interesting discussion. In the internet age, one can hardly go a single day without a new video or article being shared of "fake martial artists" being exposed, or people bashing modern-day founders for "making up their own styles." The statement that a style is made-up implies it's somehow of less value and authenticity as a traditional martial art, but those who make those accusations often study arts founded in the last 150 years.
While many Japanese ryuha often attach mystical origins to the founding of their art, such as being derived from a kami or tengu (which is why many scrolls feature them, like the one below from Shinkage Ryu), the simple truth is that every codified style of martial arts from all over the world originated with an individual. Someone accumulated a set of fighting techniques and gave it a name. To imply that someone of Asian descent is more qualified to do this than a Westerner is nothing more than a self-inflicted inferiority complex.
The last way is to be certified as a founder, but this is one of the trickiest methods. A foundership certificate cannot be issued by just anyone. It must be awarded by a legitimate Soke, the inheriting headmaster of an established martial art style. Founders cannot recognize founders, and no number of grandmaster signatures is enough. What defines any legitimate certification is the authority of the person signing it. Of course, founders can give their students permission to open their own independent branches of an art (such as when Miyagi Chojun authorized Yagi Meitoku to start the Meibukan), but that is not the same as creating a style.
So is there such a thing as a "made up style?" Not really. There are, of course, fraudulent martial artists who lie about their credentials, but when someone says a style is "made up," they are insulting the very instructors and founders of the arts they study, because the style wouldn't exist unless someone made it up.