Let $M$ and $N$ be positive dimensional, finite dimensional, smooth manifolds (with usual general topology hypotheses), and $E$ and $F$ finite rank smooth vector bundles on $M$, respectively on $N$.
Then the spaces of global sections $\mathbb E:=\Gamma(M,E)$ and $\mathbb F :=\Gamma(N,F)$ are Fréchet spaces.
I have no particular reason to doubt that $\mathbb E$ and $\mathbb F$ are isomorphic Fréchet spaces.
Is this indeed true? If yes, what is the easiest way to see this?
In the case $M=N$ and $E,F$ have the same rank, my first attempt would be as follows.
Put an auxiliary Riemannian metric on $M$ and take a cover with a countable family of geosedic balls that is trivializing for $E$. So, via an explicit local trivialization relative to $\{B_i\}, $$\mathbb E$ becomes a subspace of
$$\mathbb V:=\mathcal{C}^\infty (\coprod_i B_i,\mathbb R^{\oplus r})\simeq\bigoplus_i\mathcal{C}^\infty(B_i,\mathbb R)^{\oplus r}\;,\quad r:=\mathrm{rank}(E)\,.$$
Since, for opens $U\subseteq U' \subseteq \mathbb R^n$, restriction of smooth functions is a continuos linear map of Fréchet spaces, the above map $\mathbb E \to \mathbb V$ is a continuous linear map of Fréchet spaces as well. Since $\{B_i\}$ is a covering, the map is also injective, so an embedding of Fréchet spaces.
For every $i,j$ for which $B_i\cap B_j\neq \emptyset$, we have a map
$$\Delta_{i,j}:\mathcal{C}^\infty(B_i,\mathbb R)^{\oplus r}\oplus\mathcal{C}^\infty(B_j,\mathbb R)^{\oplus r}\to\mathcal{C}^\infty(B_i\cap B_j,\mathbb R)^{\oplus r}$$
$$(f,g)\mapsto \varphi_{ij}\cdot f|_{B_i\cap B_j}-g|_{B_i\cap B_j}$$
where $\varphi_{ij}$ is the (matrix-valued) transition function for $E$ on $B_{ij}$.
Each $\Delta_{ij}$ extends to a linear map $\mathbb V \to \mathcal{C}^\infty(B_i\cap B_j,\mathbb R)^{\oplus r}$ and to $\Delta_{ij}^k:\mathbb V\to \mathcal{A}:=\mathcal{C}^{\infty}(\cup_i B_i,\mathbb R)$ (take $k$-th component). $\mathbb V$ is a globally free rank-$r$ $\mathcal{A}$-module and each $\Delta_{ij}^k$ is a map of $\mathcal{A}$-modules. Also, $\mathbb E$ is isomorphic, as $\mathbb R$-vector space, to the intersection of the kernels of all the $\Delta_{ij}^k$.
Do the analogous construction with $F$, with transition functions $\psi_{ij}$ (assuming $\{B_i\}$ simultaneously trivializing for $F$), and get that $\mathbb F$ is $\mathbb{R}$-isomorphic to the intersection of kernels of some $\Lambda_{ij}^k$.
So, via the above embeddings we can consider $\mathbb E\subseteq \mathbb V$ and $\mathbb F\subseteq \mathbb V$.
Now, using the maps $\psi_{ij}\cdot \varphi_{ij}^{-1}$, I think one can construct an automorphism of $\mathbb V$ as an $\mathcal{A}$-module that sends $\mathbb E$ exactly onto $\mathbb F$. So, since surely an automorphism of $\mathcal{A}$-modules is an automorphism of Fréchet spaces, we have proven that $\mathbb E$ is isomorphic to $\mathbb F$.
Does the above make sense?
How to proceed if $M\neq N$ but have the same (positive) dimension? Or even if they have unequal (positive) dimension?
It's probably safer to start with
Are $\mathcal{C}^\infty(\mathbb R^n,\mathbb R)$ and $\mathcal{C}^\infty(\mathbb R^m,\mathbb R)$ isomorphic Fréchet spaces? ($n\neq m$)