If you want a canonical description of the homotopy type of the general fiber, you can take the homotopy type of the space $X\times_{\mathbb{D}}\mathbb{H}$ where $\mathbb{H}:=\{z\in\mathbb{C}\mid \Im z>0\}$ is the upper half complex plane, and the map $\mathbb{H}\to\mathbb{D}$ is the exponential map $z\mapsto e^{-z}$. This is because, as a complex manifold, $\mathbb{H}$ is the universal cover of the punctured disc $\mathbb{D}^\times=\mathbb{D}\smallsetminus\{0\}$. Since $\mathbb{H}$ is contractible and $\mathbb{H}\to\mathbb{D}^\times$ is a fibration, then the fiber product $X\times_\mathbb{D}\mathbb{H}$ deformation retracts on $X\times_{\mathbb{D}}\{t\}$ for every $t\neq 0$. The monodromy action is then given by the map $\mathbb{H}\to \mathbb{H}$ sending $z$ to $z-2\pi i$, which is a map over $\mathbb{D}$ (this is exactly the deck transformation corresponding to the standard generator of $\pi_1\mathbb{D}^\times$).

In the algebraic case you can think of $\eta=\operatorname{Spec}A\smallsetminus\{s\}$ as the algebraic analogue of $\mathbb{D}^\times$, and the generic point $\bar\eta$ as the algebraic analog of the universal cover $\mathbb{H}$. Then the Galois group of $\bar \eta/\eta$ is the one inducing the monodromy action.