That depends on local laws. Most states do not regulate instruction of any kind. But might require insurance or being bonded, etc.
Being able to teach a form of fitness whether MMA or spin isn’t dependent on a certificaiton, but being safe is something that almost always needs to be learned for teaching any class. And that is one of the best reasons to get a professional certification. Someone that is a certified fitness instructor can then acquire skills to teach other forms of training and apply their knowledge of teaching methods to the new skill.
I would not take instruction from an uncertified instructor. Many people think that having been a fighter, a swimmer, a triathlete, etc. makes them qualified to be a teacher of that material. But this isn’t always true. And it is irresponsible not to put the clients safety first and not only get safety training like CPR and First aid, but also training on appropriate methods for taking clients from a novice to an experienced exerciser in any format of fitness or athletics.
Hopefully the person you are asking this about is a safe and knowledgeable instructor even without a certification.