It's a nonsensical definition. If every creature, including PCs and harmless flies are monsters, monster has no meaning. It's like if the designers decided to define D&D as every game ever made from the beginning of time until the end of time. I mean, I guess it's a definition, but it doesn't mean anything.
For the monster definition to mean something, it needs to mean something different than PC, harmless animals, etc.
But that is a definition the game makes. Putting aside whether it's nonsensical or not (I, for one, am amused that it rightly calls the murderhobo PC's monsters), why is it there? Why does an official rulebook state that a harmless frog is a monster?
I'll attempt to answer my own question.
1) It's a joke. Haha, isn't it funny? I'm a monster! You're a monster!
Sure, D&D rulebooks are no strangers to containing humor or poking fun at themselves. But this opens the door to calling anything nonsensical in the rulebooks as a joke and not actually intended as rules. Also, the fact that it's literally in the section of "how to use this book" means this is a bad place to put a joke without specifically identifying it as being tongue in cheek.
2) It's intentionally simplistic for new DM's to understand.
That the definition takes up a sixth of the page and goes into some detail rather than just saying "you're a monster, everything is a monster, haha!" makes it sound like this is a serious attempt at a definition. And I don't think patronizing potential DM's, the people most likely to own or use this book (outside of polymorphers and Druids) is a fantastic take either.
3) It's intended to help explain why we have "Charm Person" and "Charm Monster" as spells.
As I stated before, this does line up with the methodology behind those spells, if you accept "Person" as being a subset of "Monster", and Monster containing anything one might target with the spell. But if you're trying to explain D&Disms to the player base, having that be in the Monster Manual seems fairly odd.
4) The people who wrote the Monster Manual are bad at their jobs.
I, and many others, often take the designers to task for a lot of things about the game. But I think it's completely bad faith to say that something in the book is only there because of ineptitude. And I have much better targets, like bonus action spell limitations or divine caster weapon juggling to point at if I'm trying to make a point about my dislike of their design.
In conclusion, the Monster Manual's definition of "monster" has a purpose, and just saying "Monster Manual wrong, Dungeon Master's Guide is right" and calling it a day is certainly one's prerogative to do, but I feel that line of thinking leads to a lot of other problems. If the books indeed have a hierarchy, then it basically says that there is text in rulebooks that is not only wrong, but misleading, that has never been addressed or edited.
Game rulebooks aren't meant to be sacred texts of a Gnostic religion, where "true mastery" is achieved by sifting through false statements to gain enlightenment. They're meant to show people how to play the game, and have fun by doing so. Certainly, the writers are fallible, as all humans are, but we shouldn't dismiss something as being a total non sequitur in a discussion without further evidence that this is an error.
All rulebooks are equally valid as part of the holistic rules set of the game. Certainly, newer ideas can supplant earlier ones, and anyone is free to follow or discard the contents as suits their needs. But nothing should be explicitly stated to be vestigial text. It was written with a purpose, for a reason, though we may not know what that is.
The acid test, of course, is to see what the 2025 Monster Manual has to say.