Here is a not-quite-answer. Note that $\pi_1$, but not $\pi_{>1}$, can be nonabelian. Higher homotopy groups can interact in a nonabelian, aka nonstable, way, but still they fit together solvably; $\pi_1$ can be insolvable. In terms of representation theory, whether a group is abelian translates into whether its simple representations are invertible. Indeed, for this reason physicists use the term "(non)abelian anyon" to refer to a (simple) quasiparticle which is (non)invertible under fusion, whether or not the cause is a charge under a (non)abelian group. Extended objects (quasistrings, etc.) can be nonabelian in this sense, but it seems (and it is known in sufficiently finite contexts) that any nonabelian nature is kind of fake, in the same way that solvable groups are only fake-nonabelian. Note for dimension counting: particles are 1-dimensional and strings are 2-dimensional in spacetime.
Manifolds, for example the spacetime of a quantum system, have one primary extra feature that homotopy types lack: Poincaré duality. I don't just mean this stably as a pairing on cohomology. I also mean to refer to the nonstable/nonabelian phenomenon of knotting and linking of submanifolds. The upshot is that, whereas $\pi_k X$ and $k$-dimensional bordism groups of $X$ (like $\Omega_k^O X$) are about $k$-manifolds in $X$ that may self-intersect, if $X$ is an $n$-manifold, then there are more "quantum" things like $\pi_k X$ having to do with non-self-intersecting loops or $k$-manifolds. I'm intentionally being imprecise. The point is that this more "quantum" thing has $k$ units of one type of (co)multiplication, if you are looking at $k$-manifolds in your $n$-manifold, and $(n-k-1)$ units of a different type of (co)multiplication, corresponding to the ways that $k$-manifolds can wiggle and fuse in $n$-space. Well, let me be slightly less imprecise by supplying an example. Suppose you have a 3-manifold $X$, perhaps framed (or I'll need to put some other structure somewhere else) and a braided monoidal category $\mathcal{B}$. Then you can look at the space of all skeins in $X$ labeled by objects and morphisms in $\mathcal{B}$. I want you to think that this is some sort of (space of) representation(s) of some "quantum" object that sees some of the "quantum homotopy theory" of $X$. Poincaré duality then becomes a sort of duality between $k$-manifolds and $(n-k-1)$-manifolds, given by linking them. (Or, if you want, fill your manifolds, and study intersection pairings.)
Recall that monoidal $1$-categories, but not higher (except for fake aka solvable cases), can be "nonabelian" in the physicists' sense of having noninvertibility. This supplies a richness to the skein theory of $1$-submanifolds that isn't present for higher submanifolds. On the other hand, Poincaré duality says that $1$-submanifolds are dual to $(n-2)$-submanifolds. So $(n-2)$-submanifolds also collect an extra richness that other dimensions don't have. The richest case is when $n-2 = 1$: knots and links in 3-manifolds are much richer than in any other dimension. Specifically, they have a lot more "nonabelian" nature than the general case.
I like to think of this "doubly nonabelian" nature of knots in 3-manifolds as the quantum version of insolvability of $\pi_1$. Hurewicz tells you that solvable objects with trivial homology are trivial — but there are perfect groups, and you can kill their higher homology without killing them, to produce homology-points that are not points. Pick up a manifold (with vanishing L-theory obstruction) and try to surger it to a sphere. Surgery kills homology in a Poincaré-duality-type way — it kills the abelian (quadratic) version of the "quantum homotopy" of your manifold. It can hang if your manifold has too much insolvable/nonabelian/perfect quantum homotopy theory. This only happens in specific dimensions. After more carefully counting dimensions, you find that surgery hangs in dimension 4, exactly due to nonabelian knotting in dimension 3.
There's other parts of the story I don't quite understand. For reasons I do not have a picture for, you can continue to surger at the cost of smooth structure. Specifically, Freedman supplies an interesting topological but non-smooth filling of the Poincaré 3-sphere. Recall that only the reason there even is a Poincaré 3-sphere is that there is a (super)perfect group which acts without fixed points a 3-manifold (specifically, on a 3-sphere). It is the first (super)perfect group: $A_5 = \mathrm{PSL}_2(5) = \dots$. That perfect groups exist at all is already a surprise — if you just start listing groups in increasing order, or if you randomly finite sample groups, it takes a long time to find an insolvable one. I don't really have a good story for why a perfect group can act on a 3-sphere. It requires some coincidences, the most fundamental being that there exist $a,b,c \in \mathbb{N}$ all distinct with $\frac1a + \frac1b + \frac1c = 1$. I tried to tell this story for an advanced high school audience in http://categorified.net/MathcampColloquium-novideo.pdf. And I really don't have a good story for why, at the cost of smoothness, you can evade this perfect group.
All together, I'm saying that the reason that $\mathbf{4}$-dimensional manifolds are wild is that $\mathbf{4} - 2 = 2\times \mathbf{1}$, where $\mathbf{1}$ is the categorical dimension in which perfectness (essential nonabelianness) occurs.
The existence of a 4-manifold with arbitrary $\pi_1$, on the other hand, is that $\mathbf{4} - 2 \geq 2\times \mathbf{1}$, as Ryan Budney points out in the comments. In both cases, the factor of $2$ and the shift by $2$ are about Whitney embedding statements (but they're not all the same "$2$"s).
