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I'm trying to get a better understanding of the rationale behind free groups, and more generally free objects.

This answer does a great job at explaining how various free objects are built, and I understand that one builds a set of "words", and defines an operation over this set, imposing only that a specific set of rules are satisfied.

This makes me wonder: is this construction with words really necessary, or is its only purpose to have a "concrete" object to reason with?

In other words, is studying a free (say) group equivalent to analysing what exactly can be said about a group, without attaching any specific meaning/interpretation to the group elements, so that the elements of the group are effectively only arbitrary symbols, and only the number of such symbols matters?

Along the same lines, when people say that $mathbb Z$ is a free Abelian group, is this statement effectively equivalent to be saying that $mathbb Z$ is entirely defined by its property of being an Abelian group with a single generator?

In a sense yes, studying a free object is very similar to studying the underlying equational theory : as you mentioned, a free group on one generator ($mathbb{Z}$) is the most general thing you get when you think of a group generated by one element.

But that sense is very limited, in that it seems like you want to restrict the study of an object to its equational theory. Free objects are much richer than that, and from time to time, having a concrete model for a free object (say reduced words for the free group) can sbe vert useful, even if most of the time the universal property is enough to get by.

An example that comes up way more often than one might think at first sight is the free (commutative, unital) ring on $n$ generators. One model for it is $mathbb{Z}[X_1,...,X_n]$, and this ring has (non equational) properties that are really interesting (its equational properties "aren't interesting", in that they're just the equational properties of any ring with $n$ fixed elements), for instance it's an integral domain, which allows us to use its fraction field in many arguments concerning general rings, and this comes in quite handy.

I don't know how "useful" you think that can be, but the "reduced words" model for the free group on $n$ generators allows us to prove that $F_2$ (free group on $2$ generators) contains a free group on $3$, or even infinitely many generators !

So yes, the free object on $n$ generators is "entirely defined by its property of being an object and having $n$ generators", and yes its equational theory is not more interesting than simply the equational theory you're considering; but it can have some nontrivial/interesting non equational properties that can be very useful, or at the very least interesting.

I am not a hundred percent sure this answers your question, but the Tarski problem asks whether or not the first order theory of nonabelian free groups are equivalent. This was answered in the affirmative by two groups independently: Kharlampovich-Myasnikov and Sela (spanning hundreds of pages). While proving this they also showed that there are groups with the same first order theory as free groups, but not free! As an example surface groups also satisfy the same first order theory.

So from the perspective of elementary theories you can not tell $F_2,F_3$ or $pi_1(Sigma_2)=langle a_1,b_1,a_2,b_2 mid a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1} rangle$ apart! This should tell you just first order equations is very restrictive, and there are certainly a lot more to these groups than that.

You may be interested in this mathoverflow question too.