I know part of what the book is doing -- I'm a little familiar with universal algebra.
Universal algebra is an approach that can study many diverse kinds of algebraic structures simultaneously, such as:
groups, abelian groups, rings, modules over a ring, vector spaces over a particular field, algebraic lattices, algebras over a ring, representations of a discrete group acting on a vector space over a particular field, etc.
(but not fields!)
The connecting theme is that all of these structures can be defined by writing down the allowed operations, and some equational identities they satisfy.
For example, let Omega (that should be the capital Greek letter) be the
type consisting of a binary operation *, a unary operation ^-1, and a nullary operation 1. (Yes, that's essentially just a constant, but it's fruitful to think of it as a function with no arguments)
The category of structures that have those three operations are called
Omega-algebras.
The category of groups is a
variety of Omega-algebras, defined by taking the quotient by the following relations:
1 * x = x
x * 1 = x
x * x^-1 = 1
x^-1 * x = 1
x * (y * z) = (x * y) * z
This is the general idea behind the foundations of universal algebra. Another example is vector spaces over a field K.
The type for this structure consists of the binary operation +, the unary operation -, the nullary operation 0, and one unary operation for every element of K.
Then, the equational identities include (among other things) one equation of the form
k (v + w) = k(v) + k(w)
for every element k of K.
Anyways, if you were reading what I said above, you knew logic but not algebra, you would have thought I was talking about model theory.
Specifying the underlying type of a universal algebra is essentially the same thing as specifying a formal language -- they're both simply a list of symols and how many arguments they accept.
The structures of this language are precisely the Omega-algebras.
In the univeral algebra setting, we take the quotient by certain algebraic identities to create the structures of interest.
In the logical setting, we look for structures that satisfy those algebraic identities. (i.e. models of those identities)
To repeat...
From the universal point of view, a group is nothing more than the quotient of Omega-algebra by a particular set of relations.
From the logical point of view, a group is nothing more than a structure in a particular language that satisfies a particular set of propositions.
So, foundationally at least, universal algebra is simply a special case of model theory where the statements to model take a particular form.