The Berry connection and Berry curvature only appear due to the wave nature of physical systems. That is why it also plays a role in photonics, acoustics and other classical wave equations.
The Berry connection and Berry curvature are a connection and a curvature in the mathematical sense on a vector bundle, commonly known as the Bloch bundle
\begin{align*}
\mathcal{E}_{\mathrm{Bloch}} = \bigsqcup_{k \in \mathrm{BZ}} \mathcal{H}_{\mathrm{rel}}(k) = \mathrm{span} \bigl \{ \varphi_1(k) , \ldots , \varphi_n(k) \bigr \} ,
\end{align*}
which is constructed from gluing together the eigenspaces
\begin{align*}
\mathcal{H}_{\mathrm{rel}}(k) = \mathrm{span} \bigl \{ \varphi_1(k) , \ldots , \varphi_n(k) \bigr \}
\end{align*}
spanned by the eigenfunctions associated to the eigenvalues below the characteristic energy or frequency; in solid state physics, this is the Fermi energy. Here we have assumed that your characteristic energy or frequency lies in a bulk band gap, because then the dimensionality of $\mathcal{H}_{\mathrm{rel}}(k)$ is independent of $k$ and the relevant subspace $\mathcal{H}_{\mathrm{rel}}$ of your Hilbert space depends analytically on $k$. (In classical waves, you need to pay attention to the bands with linear dispersion around $k = 0$ and $\omega = 0$, though.)
Berry connection and Berry curvature are then, as mentioned before, just a connection associated to the curvature on this vector bundle.
