This is true without any further assumptions on $M,V,W$ and discussed quite thoroughly in chapter 7 of Conlon's wonderful book "Differentiable Manifolds". The proof given in the text is quite elegant so I can't resist recalling the argument.
There is an obvious $C^{\infty}(M)$-bi-linear map $\alpha \colon \Gamma(V) \times \Gamma(W) \rightarrow \Gamma(V \otimes W)$ given by $\alpha(\xi,\eta)_p = \xi_p \otimes \eta_p$ which gives rise to a map $\alpha \colon \Gamma(V) \otimes_{C^{\infty}(M)} \Gamma(W) \rightarrow \Gamma(V \otimes W)$. We want to show that $\alpha$ is an isomorphism.
When $V$ and $W$ are trivial bundles of rank $n$ and $k$ respectively, one can choose $n$ pointwise linearly independent sections $\xi_i \in \Gamma(V)$ and $k$ pointwise linearly independent sections $\eta_j \in \Gamma(W)$ and then $\Gamma(V \otimes W)$ is easily seen to be a free $C^{\infty}(M)$ module with basis $\{ \xi_i \otimes \eta_j \}_{i,j}$ and so $\alpha$ is an isomorphism of $C^{\infty}(M)$ modules.
If $V$ and $W$ are not trivial, we interpret them as a direct summands of some trivial bundles. That is, we find vector bundles $V^{\perp}$ and $W^{\perp}$ such that $V \oplus V^{\perp}$ and $W \oplus W^{\perp}$ are trivial bundles. Then, by looking at the diagram
$$ \require{AMScd}
\begin{CD}
\Gamma((V \oplus V^{\perp}) \otimes (W \oplus W^{\perp})) @<{\tilde{\alpha}}<< \Gamma(V \oplus V^{\perp}) \otimes_{C^{\infty}(M)} \Gamma(W \oplus W^{\perp}) \\
@AAA @AAA \\
\Gamma(V \otimes W) @<{\alpha}<< \Gamma(V) \otimes_{C^{\infty}(M)} \Gamma(W)
\end{CD} $$
we see that $\tilde{\alpha}$ is injective by the previous paragraph, and we can check directly that both vertical arrows (that are defined with the help of the injective bundle maps $v \mapsto (v,0)$ of $V \rightarrow V \oplus V^{\perp}$ and $w \mapsto (w,0)$ of $W \rightarrow W \oplus W^{\perp}$) are injective, showing that $\alpha$ is injective. Similarly, by reversing the direction of the vertical arrows and replacing them with projections, we see that $\alpha$ is also surjective.
The crux of the proof is the result that every bundle $V$ over $M$ can be realized as a subbundle of a trivial bundle $F$ (stated in terms of the module of global sections, this means that $\Gamma(V)$ is a $C^{\infty}(M)$-projective module). This can be shown by constructing an epimorphism $\psi \colon F \rightarrow V$ of vector bundles from a trivial bundle $F$ onto $V$ and using a fiber metric to split $F$ as an inner direct sum $F = \ker(\psi) \oplus \ker(\psi)^{\perp}$ with $\ker(\psi)^{\perp} \cong V$.
The construction of $\psi$ uses partition of unity. When $M$ is compact, one takes a partition of unity $\{\lambda_i\}_{i=1}^n$ subordinate to a cover $\{U_i\}_{i=1}^n$ of $M$ over which $V$ trivializes with generating global sections $\xi_i^j \in \Gamma(U_i,E|_{U_i})$. Then, one can define global sections $\sigma_i^j = \lambda_i \xi_i^j$ by zero extension outside $U_i$. We obtain finitely many sections, and by taking the trivial bundle with the vector space $X = \mathrm{span} \{ \sigma_i^j \} \subseteq \Gamma(V)$ as fiber, and defining $\psi \colon M \times X \rightarrow V$ as $\psi(p,\sum \sigma_i^j) = \sigma_i^j(p)$ we obtain the required map.
Addendum:
- One can use the ideas described in the proof above to show Swan's theorem about the equivalence of categories between the category of (smooth, finite rank) vector bundles over $M$ and the category of projective finitely generated $C^{\infty}(M)$ modules.
- In the algebraic / holomorphic setting, this fails badly, at least for projective varieties. While in the smooth category, you can identify a vector bundle with the module of global sections, in other settings the module of global sections doesn't hold enough information about the vector bundle and one should consider the sheaf of sections instead. This failure also provides a geometric example of why one needs to sheafify when taking tensor products (one expects that the sheaf of sections of the tensor product will be the tensor product of the sheaves of sections and so $\Gamma(\mathcal{O}(-1) \otimes \mathcal{O}(1)) \cong \Gamma(E(1) \otimes E(-1)) \cong \Gamma(E(0)) = \mathbb{C}$ which obviously cannot hold if you don't sheafify).
- The fact that every (continuous) finite rank vector bundle is a direct summand of a trivial bundle is true even if we merely assume that $M$ is compact and Hausdorff. One can get rid of the assumption that $M$ is compact when $M$ is a manifold, but cannot get rid of it in general. In Hatcher's "Vector Bundles and K-Theory", it is shown that the tautological line bundle over $M = \mathbb{RP}^{\infty}$ is not a direct summand of a trivial bundle using characteristic classes.
Consider the following operators:
- Gradient - Acts on functions and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla \colon \Gamma(M,M \times \mathbb{R}) \rightarrow \Gamma(M,TM)$.
- Directional derivative of some vector field with respect to a fixed vector field $X$. Acts on vector fields and return vector fields. On $M = \mathbb{R}^3$, it has the signature $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
- Divergence - Acts on vector field and return functions. On $M = \mathbb{R}^3$, it has the signature $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M \times \mathbb{R})$.
They are all classical first order differential operators with different domains and codomains that can be interpreted as acting on and returning sections of vector bundles. On $M = \mathbb{R}^3$, all the vector bundles involved (and in general) are trivial and so you can just pick some global isomorphism $TM \cong M \times \mathbb{R}^3$ and think of the operators as operators with signature $D \colon C^{\infty}(\mathbb{R}^3,\mathbb{R}^n) \rightarrow C^{\infty}(\mathbb{R}^3,\mathbb{R}^m)$ for appropriate $n$ and $m$ (they act on vector valued functions and return vector valued functions). However, working with arbitrary manifolds, the notion of a vector field which was before (identified with) just a tuple of smooth function (an element of $C^{\infty}(\mathbb{R}^3,\mathbb{R}^3)$) now generalizes to a section of a possibly non-trivial vector bundle $TM$ and so the operators now should be generalized to operators that act on sections of (possibly non-trivial) vector bundles. And indeed,
- Gradient is replaced/generalized with the differential of a function $d \colon \Gamma(M, M \times \mathbb{R}) \rightarrow \Gamma(M, T^{*}M)$ (function $\mapsto$ covector field) or, in the presence of a Riemannian metric, by a gradient operator $\nabla \colon \Gamma(M,M\times \mathbb{R}) \rightarrow \Gamma(M,TM)$ (function $\mapsto$ vector field).
- The directional derivative is generalized by the notion of a covariant derivative (also called an affine connection) denoted by $\nabla_X \colon \Gamma(M,TM) \rightarrow \Gamma(M,TM)$.
- Divergence is generalized in the presence of a Riemannian metric to an operator $\mathrm{div} \colon \Gamma(M,TM) \rightarrow \Gamma(M,M\times \mathbb{R})$.
Best Answer
Here's one argument using bump functions.
Let $\beta,\gamma:\Gamma(U,E)\to\Gamma(U,F)$ be two such restrictions. It suffices to show that $\beta-\gamma=0$.
Choose any local section $s\in\Gamma(U,E)$, and any $x\in U$. It now suffices to show that $(\beta-\gamma)s(x)=0$.
We may always find a compactly supported bump function $\psi:M\to\mathbb{R}$ such that $\psi=1$ on an open neighborhood $V\ni x$ and $\text{supp}(\psi)\subset U$. We know by locality that $(\beta-\gamma)s(x)=(\beta-\gamma)(\psi s)(x)$ (since $s-\psi s$ vanishes on $V$), and since $\psi s$ may be smoothly extended to a section $\widetilde{\psi s}$ on all of $M$, we have $$ (\beta-\gamma)s(x)=(\beta-\gamma)(\psi s)(x)=\beta(\psi s)(x)-\gamma(\psi s)(x)=\alpha(\widetilde{\psi s})(x)-\alpha(\widetilde{\psi s})(x)=0 $$
Edit:
As probably123 points out, this proof shows that the restriction $\alpha|_U$ is unique among local operators. If there is no requirement that the restriction be local, then there is no guarantee of uniqueness.
The space of restricted sections $\Gamma(M,E)|_{U}:=\{s|_U:s\in\Gamma(M,E)\}$ is a proper subspace of $\Gamma(U,E)$ whenever there exist local sections that cannot be smoothly extended. This means that the quotient $Q:=\Gamma(U,E)/\Gamma(M,E)|_U$, as well as its algebraic dual $Q^*$, are nontrivial.
In this case, we may choose a nonvanishing section $t\in\Gamma(U,F)$ and a nonzero element $\lambda\in Q^*$. Define a local operator $\psi:\Gamma(U,E)\to\Gamma(U,F)$ by $\psi(s)=\lambda([s])t$ (where $[\ ]$ denotes the projection into $Q$). Since $\psi$ vanishes on $\Gamma(M,E)|_U$, for any restriction $\alpha|_U$, $\alpha|_U+\psi$ is also a valid and distinct restriction.
A more explicit counterexample is hard to find in the smooth category since $Q^*$ is rather difficult to describe. It may also be possible to ensure uniqueness in other ways, such as topologizing the spaces of sections and requiring continuity, or restricting attention to compactly supported sections.