I don't know the specifics of what that book says, in regards to a "tri library". I don't own that book, nor have I read it. It's possible somebody else around here has, though.
The XNA templates include an option that's like a "game library". A library that can be reused across multiple games, but isn't a game itself. It compiles down to a DLL, which you can reference in your main game project and reuse. It kind of sounds like that is what the book is directing you to do.
MonoGame, on the other hand, does not have a template for a game library. I suspect this is your problem, or at least a significant part of it.
But that's not to say that you can't create a project on your own that works in this same way.
The only thing that made the XNA Game Library project type really any different from the Class Library project type is that it was already referencing the XNA libraries.
You ought to be able to get a similar result to what I think the book is asking for by doing something like this:
1. Create a new project using the Class Library project type/template. It's not specific to XNA or MonoGame. You should be able to find it under the Visual C# category, rather than under MonoGame or XNA Game Studio.
2. Take that project and add references to the MonoGame libraries. Off the top of my head, I don't know how many there are or what their specific names are, though I think there might only be one. You can figure this out by going to your main game project in the Solution Explorer and inside of your game's project, open up the References node. Look through that list to find anything that mentions MonoGame. There will be at least one, but I'm thinking there might be only one. The trick then, is to go to your new class library project and right-click on it and add references to those same MonoGame projects so that they match. Once that is done, you will be able to create classes inside of the new class library project that use MonoGame code (or XNA style code, since the API is the same between the two).
3. You'll also have to add a reference in your main game project to your new class library project in order for your main game to utilize the code in it.
I know those steps are somewhat vague. I don't have Visual Studio open right now to do further digging to provide specifics. But I'm hoping it at least breaks down your problem into some smaller problems that you can begin to work your way through.
Again, maybe somebody else around here owns that book and has worked their way through it and can provide more specific insight. But I didn't want your question to go without any sort of answer for too long.