This is a good question. To paraphrase, are there official / definitive kata bunkai, or are people free to interpret their kata however they like?
The answer is: The latter, mostly.
First, Okinawan karate originally came from Chinese kung-fu. That much we are 100% sure of. And Japanese karate derived from Okinawan karate. Of course, there were lots of other influences, such as classical Japanese jujitsu on Japanese karate. And Okinawan karate had been influenced by Okinawan wrestling styles which themselves may have come from other nations. But the transmission from Chinese kung-fu to Okinawan karate to Japanese karate is the main thing to understand.
The problem is, whenever a martial art is passed down from teacher to student, the transmission can be incomplete. As was often the case, the full Chinese kung-fu art generally would not have been passed on. That full transmission was often limited to the family teaching the art. It was a way of ensuring that the family continues to make money.
Forms are easy to learn, but hard to understand without being taught the meaning behind the movements. And typically, that was where all of the knowledge was. The movement is cheap. Anyone can learn that just by observation. But the application and meaning behind it is going to be taught only to those who could afford it, and usually the family members withheld some of the more advanced or practical interpretations for themselves while publicly teaching the less practical or more basic applications. At least that was the case for Chinese kung-fu.
As for Okinawan karate, not much is really known about how much of each style was truly transmitted to each of the original founders of Okinawan karate. We do know that different forms (kata) would come from multiple different teachers of potentially many different styles over hundreds of years. So karate has a lot of different original teachers. It's a blend of different ideas and styles.
What's more, it's common in Okinawan karate that the complete art was only passed down from teacher to one or two students who will then go on to lead the art for the next generation. This is what I've been able to gather over the years. It's not discussed much. But the idea is that maybe only one or two people of high moral character and dedication to the art would be trusted enough by the master to pass down the complete art to them.
As a result, a lot of the "official" bunkai are lost. There simply aren't enough people being taught it from generation to generation for the bunkai to survive to modern day.
And pretty much all karate masters today do not specify an "official" list of kata bunkai for each movement. I've always said that they really need to do this. Instead, it's left to the student to study and figure it out on their own. And the tradition of gaining the trust of a master enough for him to show you his bunkai lives on.
Mind you, it's not that hard to figure it out. Form and function go hand in hand. If you study something like classical Japanese jujitsu, you will be able to just see it. That's because that's what it is, basically. It's all classical jujitsu-like technique, with maybe stylistic differences.
Well, let me walk that back a bit. What makes kata bunkai harder is the fact that the kata are not the same as the original kata. Instead, kata change over time. Errors creep in. It's like playing the "telephone game". Techniques that seemed strange and maybe even "weak" were often changed into techniques that made more sense or seemed stronger to those who simply had no idea what the strange looking techniques actually were. And those teachers passed it on to their students who did the same. Today, we have examples of different branches of karate doing the same kata differently. It certainly happened.
And so, if the kata today are not exactly the same as the kata as they were originally performed, then all bets are off. There are karate historians who look at the differences in the kata over time in an attempt to piece together what the original kata must have looked like. And they go all the way back to China, looking at the original kung-fu arts that karate was based on. And I have to tell you, some of the basics remain intact in karate. It's recognizably from, say, White Crane kung-fu or Incense Shop Boxing. But there are often many details that are changed. Some karate kata represent only the first half of the original Chinese kung-fu form, for example. The missing half was simply dropped from karate, or maybe it was never taught to whoever it was that brought it into Okinawan karate.
But, if you take a very simplistic approach and just look at the kata as they are today, you can still derive bunkai using a knowledge of classical jujitsu. Some techniques done in kata can only be one possible thing in jujitsu. Others are more open to multiple possible interpretations.
There are some bunkai that are taught by the current day heads of each branch of karate. Those seem "official" enough. But even then, it's usually never stated as such, which does make it confusing.
With regards to how to know if a bunkai interpretation is "right" or "wrong", this is a grey area, as you might suspect. What does it mean to be right or wrong when it's completely subjective?
The answer is that it's not completely subjective. There are bunkai that work when pressure tested, and there are bunkai that don't. There are bunkai that are too basic to believe might be a candidate for the original bunkai. For example down-blocking a low kick. You learn to block from sparring, not from kata. The original kata, one presumes, would be designed to contain only techniques that were not obvious. They could not be learned from sparring. These would need to be cool enough to want to encapsulate and preserve in a kata for all eternity. A down-block to a kick probably wouldn't be high on the list of things to want to pass down.
So kata bunkai can be "wrong". In the end, it's up to the individual student to learn better bunkai for the same technique and then discard bunkai that seem wrong. And after a good amount of experience, they all know the difference when they see it.
As for Shotokan karate, I go over this more at my answer at the following link: Styles of Karate
In short, kata bunkai simply wasn't taught in Shotokan originally. It wasn't passed on, because it was taught merely as a form of physical exercise to school children in Japan. The more violent aspects had to be suppressed. They didn't want kids going around seriously hurting each other with it. So kata bunkai was not taught to most of the original Japanese shotokan students. In modern days, people in Shotokan karate are actually learning it through the internet, videos, and Okinawan karate instructors.
That's all I wanted to say on it. I wish I had some more authoritative references to give you. Most of that is from me reading articles and books about it over time. If you really want to know more, I'd refer you to a well known man, Patrick McCarthy, who is an Okinawan karate historian. He has many books and articles on this subject. I think everyone looking to get into the history and origins of karate would do well to read what he has to say.
Hope that helps.