The book seems to consider it such obviously basic knowledge that it's not worth explaining, but a moderately attentive reading of the work and the examples it chooses for its explanations will make it fairly clear that this book is primarily concerned with people who seek to become Pokémon trainers, which as described in the book appears to be a particularly popular sport that also segues into a few related careers like ranger, police officer, field research, and gym trainers, whatever those are. There are no examples from the military, but whether that's because this is a kids book, because wars are different here, or if they have a genuinely different career path is not particularly obvious.
According to the book, the reason is complex politics, although it will gloss over many of the details!
A few hundred years ago, most countries didn't have any limitation on the number of Pokémon! This was because up until about 500 years ago, having more than one or two pokemon that you personally befreinded was basically unfeasable due to the primativety of apricorn carving (the earliest forms of Pokéballs) at the time and lack of modern food supplies. As such, they were largely unprepared for the troubles of the time, where people would capture and imprison whole groups of pokemon, and force them to do their bidding. The nobles of many nations would often have entire armies at their beck and call.
However, multinational cooperations between ethical and practical reform groups ended up changing that. It was horrible for the Pokémon involved, and contributed to the ability of evil elites to opress the average citizens. They rolled out the reforms by vote and law in willing countries, and soon the limits spread to the entire world. While some groups resisted for a while, even rare and powerful Pokémon are less powerful than normal ones if the latter are willingly cooperating and willingly trained, not to mention being more difficult to obtain.
The limit of 6 Pokémon isn't a universal one, but it's standard for most of the world, including all of America. Setting the limit at 6 Pokémon helps ensure that people don't capture more pokemon than they can care for and get along with. Most people, even including many trainers, won't ever reach this limit, and instead focus on working to help few of their best freinds draw out all of their potential!
It scrupulously avoids going far into details, but this case it seems far more clearly to be because of the intended age range (7-10, the cover helpfully adds).