We study some anisotropic equations involving the Finsler $N$-Laplace operator $ \Delta_{H, N},$ with Trudinger-Moser type critical exponential nonlinearity which involves the anisotropic norm. More precisely, we establish the existence of nontrivial solutions to the problem
\begin{equation}
-\Delta_{H, N} u = f(x, u),\;u\geq0,
\end{equation}
on a smooth bounded domain $ \Omega \subset \mathbb{R}^N, N \ge 2$, and to the anisotropic quasilinear Schr\{o}dinger type problem
\begin{equation}
-\Delta_{H, N} u + V(x) |u|^{N-2} u= g(x, u),\; u\geq0,
\end{equation}
in the whole Euclidean space $\mathbb{R}^N.$ Here $\Delta_{H, N} u = \text{div} (H(\nabla u)^{N-1} \nabla_{\xi} H(\nabla u))$ is known as Finsler $N$--Laplacian or anisotropic $N$--Laplacian, $ H: { \mathbb{R}^N} \to [0, \infty)$ is a positively homogeneous Minkowski norm, and $\nabla_{\xi}$ denotes the gradient operator with respect to the $\xi$ variable. The nonlinearities $f$ and $g$ exhibit critical exponential type growth, and the potential $V$ is a continuous function that retains the compactness of the associated Sobolev embedding into a larger class of Lebesgue spaces under suitable assumptions.