Any topological group has an abelian fundamental group by the Eckmann-Hilton argument. Is there some intuition behind the fundamental group being abelian that would enable one to predict this beforehand?
In general, abelian-nes of a group tends to convert it into a module over $\mathbb{Z}$ and this explains stuff like the Classification of fundamental groups. Is there something similar that happens when we consider abelian fundamental group?
I would just like some heuristics to think about abelian fundamental groups with.
Your first paragraph is in some sense backwards: you should hurry to incorporate the Eckmann-Hilton argument into your intuition, and then you can use it to predict lots of other things, including but not limited to the fact that topological groups have abelian fundamental groups! For example, you can also use it to predict that the higher homotopy groups $\pi_n, n \ge 2$ are abelian, or that the group of automorphisms of the identity functor on a category is abelian.
Here are some general but somewhat incorrect definitions. An $E_1$-algebra is a monoid object. If we've defined an $E_n$-algebra already, then inductively an $E_{n+1}$-algebra is a monoid in $E_n$-algebras. In other words, an $E_n$-algebra is an object equipped with $n$ compatible monoid structures.
Now, in ordinary categories this definition turns out to be very boring: by the Eckmann-Hilton argument, if $n \ge 2$ then all of the monoid structures must agree and be commutative. So for $n \ge 2$ it seems that $E_n$-algebras are just commutative monoids.
However, in higher categories things can be much more interesting, and in particular $E_n$-algebras define a different concept for each value of $n$. In this setting the correct definition is instead that an $E_1$-algebra is a "homotopy coherent" monoid; it is not required to be associative on the nose but only to be associative up to "coherent homotopy." Historically the most important example is that the based loop space $\Omega X$ of any topological space is an $E_1$-algebra in spaces, even though it is not, say, a topological group on the nose. More generally, the $n$-fold loop space $\Omega^n X$ of any topological space is an $E_n$-algebra in spaces.
You can also talk about $E_n$-algebras in categories. Here an $E_1$-algebra is a monoidal category, an $E_2$-algebra is a braided monoidal category, and an $E_n$-algebra, $n \ge 3$, is a symmetric monoidal category. So things stabilize after three steps instead of two. Braided monoidal categories are important, among other things, in descriptions of certain kinds of knot invariants.
Anyway, the upshot of all of this is that $E_n$-algebras are a natural source of commutative monoids, as follows:
One such functor is the connected components functor $\pi_0$ from spaces to sets. The relevance of this to topological groups is that topological groups $G$ are naturally $E_1$-algebras, so loop spaces $\Omega G$ of topological groups are naturally $E_2$-algebras, and $\pi_0(\Omega G) \cong \pi_1(G)$.
But the real upshot is that the $E_2$-algebra structure on $\Omega G$ is much richer than just the abelian group structure on its $\pi_0$; for example, it equips the homology $H_{\bullet}(\Omega G)$ with the structure of a Gerstenhaber algebra (up to some degree conventions).