I have revised and reposted this question here.
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 $V_\text{p}(\mathcal{M}):=(S(\mathcal{M}),+_\text{p},\cdot_\text{p})$ be the vector space of smooth functions on $\mathcal{M}$ where $+_\text{p}$ and $\cdot_\text{p}$ are the point-wise addition and scalar multiplication operations on $\mathcal{M}$. (Edit: As Wofsey rightly notes below, we could alternately have here any pointwise operations, $\oplus_\text{p}$ and $\odot_\text{p}$, for any vector space structure $(\mathbb{R},\oplus,\odot)$ for $\mathbb{R}$.)
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 $V_\text{p}(\mathcal{M})$. Namely, it distributes over $+_\text{p}$ and $\cdot_\text{p}$ as follows:
$d^*(\alpha \cdot_\text{p} f +_\text{p} \beta \cdot_\text{p} g)=\alpha \cdot_\text{p} d^*f +_\text{p} \beta \cdot_\text{p} d^*g$
for all $\alpha,\beta\in\mathbb{R}$ and all $f,g\in S(\mathcal{M})$. (Edit: Again, as Wofsey notes this holds true also for every point-wise operation $\oplus_\text{p}$ and $\odot_\text{p}$.)
My original question was as follows: Could there be some other vector space structure $V_\text{alt}(\mathcal{M}):=(S(\mathcal{M}),+_\text{alt},\cdot_\text{alt})$ for $S(\mathcal{M})$ such that every $d^*$ is linear with respect to $V_\text{alt}(\mathcal{M})$ as well as $V_\text{p}(\mathcal{M})$?
As Wofsey correctly notes below, we could here have $V_\text{alt}(\mathcal{M})=(S(\mathcal{M}),\oplus_\text{p},\odot_\text{p})$ for any of the point-wise operations mentioned above, $\oplus_\text{p}$ and $\odot_\text{p}$. My revised question is, could there be some vector space structure $V_\text{alt}(\mathcal{M})$ other than these? Namely, given that every $d^*$ is linear, could the vector space structure be anything other than pointwise?
Note that the maps $d^*$ more generally preserve any pointwise-defined operations. So, they will preserve any vector space structure that is defined pointwise, not necessarily by the standard operations. That is, let $\oplus,\odot$ be the addition and scalar multiplications of any $\mathbb{R}$-vector space structure on the set $\mathbb{R}$. Then $S(M)$ becomes an $\mathbb{R}$-vector space with operations $\oplus_p$ and $\odot_p$ defined by $(f\oplus_p g)(x)=f(x)\oplus g(x)$ and $(c\odot_p f)(x)=c\odot f(x)$. For any diffeomorphism $d:M\to M$, $d^*:S(M)\to S(M)$ is then linear with respect to these operations.