Aren't unit cell and the relevant crystal similar?
The statement from your book is correct, though the devil lies in the details. The point here is that the book is treating a theoretical crystal (i.e. one that is infinitely large). Let me elaborate and then answer your question.

A unit cell is simply a representation of a periodic structure. It is a means of conveying translational symmetry (and perhaps other types of symmetry as well, but let us focus on translational symmetry). If you have for instance a cubic unit cell, than a 3x3x3 grid of these unit cells is also a unit cell of that same lattice. If you extrapolate this idea, then the reasoning you pose in your question makes a lot of sense!

Let us now discuss the details. Assume that we have a theoretical crystal of infinite dimensions. What is then the unit cell? Can we define a unit cell which is also the infinite crystal? Every finite unit cell would obviously not be that crystal. This explains why we even use a unit cell in the first place: By defining the translational symmetry by means of the unit cell, we can define an infinite crystal.

In the field of solid state physics, it is attempted to keep things simple and well-defined and therefore a so-called primitive unit cell is introduced. This unit cell only contains one lattice point and you could see this as the smallest unit cell possible that is still able to convey the translational symmetry of the infinite crystal.

Your last question relates to why we want to use, for instance, the primitive unit cell or some other unit cell rather than the complete crystal? Let me answer this question by giving an example: If you cut a crystal, you will create a surface facet. Such surfaces also have translational symmetry and you can define these using Miller indices. Your textbook will probably treat these as well. Those Miller indices are defined by the reciprocal lattice vectors of your unit cell. You already touched upon on the fact that there exists an infinite set of unit cells that define that infinite crystal, yet the question arises which unit cell to use if we, for instance, want to use these Miller indices. In principle, any unit cell in the set would be valid. To avoid any kind of ambiguity, we tend to use the smallest unit cell possible.

A nice book treating these concepts is "Introduction to Solid State Physics" from Charles Kittel.