This is exactly the "specification by universal property" that the author is talking about. In your case, it's saying "a 3-dimensional vector space is a vector space with three chosen vectors e, f, g and three linear maps x, y, z such that each vector v equals x(v) e + y(v) f + z(v) g". But as the author points out, not everything follows from universal properties, and when it does, there is often several universal properties defining the same object, and that sameness itself needs to be shown.