This is a revision of my earlier question.
Let $\mathcal{M}$ be a finite-dimensional smooth manifold. Let $S(\mathcal{M})$ be the set of all smooth real-valued functions, $f:\mathcal{M}\to\mathbb{R}$. For any $f\in S(\mathcal{M})$ and any diffeomorphism, $d\in\text{Diff}(\mathcal{M})$, let $d^*f:=f\circ d^{-1}$ be the lifted action of $d$ on $S(\mathcal{M})$. Finally, let $V_\text{p}(\mathcal{M}):=(S(\mathcal{M}),\oplus_\text{p},\odot_\text{p})$ be a vector space of smooth functions on $\mathcal{M}$ where $\oplus_\text{p}$ and $\odot_\text{p}$ are some point-wise addition and scalar multiplication operations on $\mathcal{M}$. Namely, we could here have any pointwise operations,
$(f\oplus_\text{p}g)(x):=f(x)\oplus g(x)$ and $(\alpha\odot_\text{p}f)(x):=\alpha\odot f(x)$
where $\oplus$ and $\odot$ grant $\mathbb{R}$ a vector space structure as $(\mathbb{R},\oplus,\odot)$. The usual addition and multiplication (namely + and $\cdot$) would work here but there are other possibilities as well.
One can easily show that for any diffeomorphism, $d:\mathcal{M}\to\mathcal{M}$, its lifted action, $d^*:S(\mathcal{M})\to S(\mathcal{M})$, is linear with respect to every such vector space $V_\text{p}(\mathcal{M})$. Namely, it distributes over $\oplus_\text{p}$ and $\odot_\text{p}$ as follows:
$d^*(\alpha \odot_\text{p} f \oplus_\text{p} \beta \odot_\text{p} g)=\alpha \odot_\text{p} d^*f \oplus_\text{p} \beta \odot_\text{p} d^*g$
for all $\alpha,\beta\in\mathbb{R}$ and all $f,g\in S(\mathcal{M})$.
My question is as follows: Could there be some other vector space structures $V_\text{alt}(\mathcal{M}):=(S(\mathcal{M}),+_\text{alt},\cdot_\text{alt})$ such that every $d^*$ is linear with respect to $V_\text{alt}(\mathcal{M})$ as well as to all of the point-wise vector spaces $V_\text{p}(\mathcal{M})$?
Phrased differently, my question as follows. Suppose that you are given some unknown vector space $V(\mathcal{M})$ (i.e., either some $V_\text{p}(\mathcal{M})$ or some $V_\text{alt}(\mathcal{M})$) and told that every lifted diffeomorphism, $d^*$, is linear with respect to $V(\mathcal{M})$. Could the vector space structure of $V(\mathcal{M})$ be anything other than pointwise?
Let $G$ be the group of diffeomorphisms $M\to M$ and consider $S(M)$ as a $G$-set. So, you are asking about vector space structures on $S(M)$ which are compatible with the $G$-set structure, in that they make $S(M)$ into an $\mathbb{R}[G]$-module. Given any automorphism $F:S(M)\to S(M)$ as a $G$-set, you can transport any compatible vector space structure along $F$ to get another compatible vector space structure.
But such automorphisms are easy to find, since the structure of a $G$-set is very simple: any $G$-set is just a disjoint union of orbits. So for instance, all the constant functions on $M$ are singleton orbits, and so there is an automorphism $F:S(M)\to S(M)$ that permutes the constant functions according to an arbitrary bijection but is the identity on all the non-constant functions. If you start with the standard vector space structure on $S(M)$ and transport it along such an $F$, you will typically get a compatible vector space structure that is not defined pointwise. For instance, if $M$ has two points and $F$ swaps the $0$ and $1$ constant functions, then the resulting vector space structure on $S(M)=\mathbb{R}^2$ has $(0,1)+(1,1)=(0,1)$ but $(0,1)+(1,2)=(1,3)$ so it cannot be defined pointwise since you would need to have both $0+1=0$ and $0+1=1$.