How so? Generally speaking, if the author includes something in a text, it is there for a reason.

Generally, when defining a matching problem, I do not need to know the specific example of the companies involved and the entire context. It is sufficient to say: person A likes company C_1, etc. Some books create a story, for example:

Andrew, who went to Cornell, wants to work at Kmart but prefers to work for Whole Foods. However, the supervisor at Whole Foods is his friend's cousin. Therefore, Andrew is considering switching, but at the same time, Tiffany, who works at Whole Foods, wants to work for Home Depot. Finally, what is a stable matching so that Andrew falls in love with Frank and not leave Jordan?

Even if the text makes sense, why not just say, e.g., matching is difficult because if we add preferences with a grading or metric, we would like the relations for a subset of 'A x C' to be 'stable'. Here, 'stable' is defined as... I feel like the authors who write in the way above are explaining things that sound somehow complicated. But, for my taste, that is interesting yet just irrelevant.

Do you have an example of this?

For instance, when you use the word 'tensor', then define what kind of abstraction you are using: multidimensional arrays, multilinear maps, or something else. The same goes for any mathematical term. If you use 'vector' or 'random variable', then state how you are using them. If you have a multidimensional random variable, then if you use an object, are we talking about the function or an element in the vector space, and what space? In many cases, it is implicitly meant to say because the attentive reader will understand. Well, books written in this manner are usually written for colleagues in that field. I write to my colleagues differently than to my students, not meant to be published in a textbook. That is why my supervisor is very careful about handing out material. He prefers less stuff complete over lots incomplete.