The simple truth is yes, certifications and licenses matter. This isn't a concept unique to martial arts. A barber can't cut your hair without being licensed. You can't sell your home, paint your nails, vent about your problems or learn the alphabet without someone being licensed (at least not professionally). Not only does someone need to have the ability to perform these tasks, but they also need the training and credentials because they help reassure the quality and validity of the work being done.
In the context of martial arts, certifications help protect a line of transmission from the source. When it comes to techniques and training methods designed to increase your survivability in combat, ensuring you're learning the right material becomes quite important. On the flip side, styles and systems are often the culmination of an individual's life's work and ensuring they are being carried on properly is of equal importance. While the average student is probably not charging into battle relying on the techniques they learned in class to make it home, regular training is a physically demanding endeavor that itself can lead to injury or worse. Being able to ensure the safety of our students, along with everything else we deal with as instructors, is essential.
Of course, in that statement I'm talking about instructor licenses specifically. But what about those who have no interest in teaching? Can someone just train without worrying about certificates? Absolutely. But if they are being issued certificates, they need to be valid...
Essentially, the instructor issuing the certificate needs to have the explicit authority to do so. Otherwise, it is fraudulent. Unless they have started their own system or organization that they are the head of, the art they teach is under someone else's authority. No matter how good they are, how much you like the classes, or any number of excuses you can come up with, it's a simple matter of permission. Anyone can share whatever information they like, but if you want the authority to recognize another person officially, you need the permission to do so. Uncle Billy could be the best driver in the world, but if you simply learn from him and he writes you a permission slip to drive on a 3x5 index card, and you do not seek a proper drivers license, your ability to operate a motor vehicle on government roads is not valid.
Take for example the certification pictured here from the first time I took a pistol safety course. Although the instructor was an NRA certified instructor, he specifically had to state even on the certificate itself that the class itself was a "Non-NRA Class" because it did not meet the minimum standards to be classified as such. Otherwise, it would be a fraudulent certificate. This is just like in the Asian martial arts. You can teach whatever you want but if you want to issue certifications under the name of the style or organization you belong to, you as the instructor need to meet the standards in order to do so and your students need to meet the standards to receive them.
Unfortunately, here in the States just about anyone with a credit card can go online and buy a uniform and black belt and begin teaching. If they have any modicum of skill, the general public is none-the-wiser... And to be honest, many simply don't care. That being said, as the torch bearers of a legacy we can be proud of, classical martial artists need to continue making an effort to educate those around them. That is only way we can affect real change.