What is your personal definition of the term "unit vector?" Do you suppose there's a reason why no other textbook or Web site defines it the way these authors do?

Are there only N unit N-vectors, as the book says, or are there an infinite number of them?

My definition of unit vectors is the same as yours. However, the book does not say there are only N unit N-vectors. It says there are only N STANDARD unit N-vectors.

In math, “Adjective X” usually means something more specific than “X”. “Prime numbers” are a subset of “numbers”, and so on. Just like in this case, “standard unit vectors” are a subset of “unit vectors”.

So why don't they tell readers what unit vectors actually are, in the general case? It's a rather important elementary concept, isn't it?

Why bother describing a specific case using ambiguous language (the parentheses) and omit the one property that actually makes a unit vector a unit vector?

Bad writing in a math textbook is a weird thing to defend.