But this technique is not limited to just karate, it's used in multiple martial arts and from what I've heard it's called a 'kick up' or something along those lines.
Most martial arts do not focus on this kind of technique though they practice it as kind of a bonus technique and is not included in testing and grading.
This is not a martial arts technique. It is a kip up. I learned to do this as a young child. I saw it in a Bruce Lee movie. Like mos kids I was trying to imitate what I saw. My gym teacher saw me trying while we were on the mat learning how to do rolls for gymnastics. He gave me a couple of pointers ad before you know it I was able to do them easily. Later I taught myself to do them without using my hands. Later in high school in gym class again we were n gymnastics. Our teacher had us doing them. There were only 2 guys in the class that knew how before the teach began to teach it. Both of us could tumble. He was a much better tumbler than I. On out test he gave some bonus point for doing it without hands. I got those points. The other guy had to try several times before he was able to do it. No one else in the class got those bonus points. A few tried.
The kip up is dome in gymnastics and in the movies. This is not something that most people would ever think of doing in self defense. Capoeira people may used it in their art.