Self adjoint operators enter QM, described in complex Hilbert spaces, through two logically distinct ways. This leads to a corresponding pair of meanings of the commutator.
The former way is in common with the two other possible Hilbert space formulations (real and quaternionic one): Self-adjoint operators describe observables.
Two observables can be compatible or incompatible, in the sense that they can or cannot be measured simultaneously (corresponding measurements disturb each other when looking at the outcomes). Up to some mathematical technicalities, the commutator is a measure of incompatibility, in view of the generalizations of Heisenberg principle you mention in your question. Roughly speaking, the more the commutator is different form $0$, the more the observables are mutually incompatible. (Think of inequalities like $\Delta A_\psi \Delta B_\psi \geq \frac{1}{2} |\langle \psi | [A,B] \psi\rangle|$. It prevents the existence of a common eigenvector $\psi$ of $A$ and $B$ - the observables are simultaneously defined - since such an eigenvector would verify $\Delta A_\psi =\Delta B_\psi =0$.)
The other way self-adjoint operators enter the formalism of QM (here real and quaternionic versions differ from the complex case) regards the mathematical description of continuous symmetries. In fact, they appear to be generators of unitary groups representing (strongly continuous) physical transformations of the physical system. Such a continuous transformation is represented by a unitary one-parameter group $\mathbb R \ni a \mapsto U_a$. A celebrated theorem by Stone indeed establishes that $U_a = e^{iaA}$ for a unique self-adjoint operator $A$ and all reals $a$. This approach to describe continuous transformations leads to the quantum version of Noether theorem just in view of the (distinct!) fact that $A$ also is an observable.
The action of a symmetry group $U_a$ on an observable $B$ is made explicit by the well-known formula in Heisenberg picture:
$$B_a := U^\dagger_a B U_a$$
For instance, if $U_a$ describes rotations of the angle $a$ around the $z$ axis, $B_a$ is the analog of the observable $B$ measured with physical instruments rotated of $a$ around $z$.
The commutator here is a first-order evaluation of the action of the transformation on the observable $B$, since (again up to mathematical subtleties especially regarding domains):
$$B_a = B -ia [A,B] +O(a^2) \:.$$
Usually, information encompassed in commutation relations is very deep. When dealing with Lie groups of symmetries, it permits to reconstruct the whole representation (there is a wonderful theory by Nelson on this fundamental topic) under some quite mild mathematical hypotheses. Therefore commutators play a crucial role in the analysis of symmetries.
Best Answer
There's more to it, and the deeper content it encodes is related to symmetries and their associated conserved quantities. Let us start with the classical theory, to see that this is indeed already present there. In Classical Mechanics we may formulate our theory in the Hamiltonian Formalism. In that case we have a phase space $(\Gamma,\Omega)$ where $\Gamma$ is a space, which in basic mechanics courses is usually described as the space of pairs $(q^i,p_i)$ of position and momenta, and where $\Omega$ is an object called sympletic form.
The sympletic form gives rise to an operation among functions on $\Gamma$ called the Poisson bracket $\{,\}$. The Poisson bracket between position and momenta obey $$\{q^i,p_j\}=\delta^i_{\phantom i j}\tag{1}\label{ccr}.$$
Now, you might be aware of a result known as Noether's theorem which puts in correspondence symmetries and conservation laws. In the Hamiltonian Formalism it can be phrased as follows. For a given symmetry we have a function in $\Gamma$, called its Hamiltonian charge $Q$, which has the property that $$\{Q,f\}=-\delta_Q f\tag{2}$$
where $\delta_Q$ is the variation of the observable according to the symmetry corresponding to $Q$.
Now let us consider translations. Consider a translation by $\epsilon^i$ so that the coordinates get transformed as $q^i\to q^i+\epsilon^i$. We will have $\delta q^i = \epsilon^i$. In that regard, observe that if we define $Q = \epsilon^i p_i$ we have $$\{ Q,q^i\}=\epsilon^j\{p_j,q^i\}=-\epsilon^i = -\delta_Q q^i\tag{3}.$$
Observe that (1) has been used in the second equality. What this tells is that (1) is the statement that momentum is the generator of translations, or else that momentum is the Hamiltonian charge associated to translations. In particular, momentum in the $i$-th direction generates translations in the $i$-th direction, that is the content of (1).
This then naturally generalizes to Quantum Mechanics. And it is not so surprising that it happens, since we know that the correspondence principle gives the quantization rule $[] \leftrightarrow i\{\}$. In that setting, the Canonical Commutation Relations are just saying that momentum should be the generator of translations.
Obviously, the whole analysis of symmetries that I have outlined above in Classical Mechanics can be made in a self-contained manner in Quantum Mechanics. I only did it in Classical Mechanics to show you that there is a classical version of the story, which may be easier to understand first.
In summary, commutation relations often encode symmetry statements and their associated charges, and the CCR is just one example of that.