Let $V$ be an even-dimensional euclidean vector space and $\mathbb{C}\mathrm{l}(V)$ the complexification of the Clifford algebra, then there exists a spinor module, i.e. a pair $(S,c)$ consisting of a complex vector space $S$ and an isomorphism $c\in\mathrm{Hom}(\mathbb{C}\mathrm{l}(V),\mathrm{End}(S))$ of complex algebras - in other words, $(S,c)$ is a Clifford module such that $c$ is invertible. See proposition 3.19 in Heat Kernels and Dirac Operators. The proposition also says that the spinor module is unique up to an isomorphism of Clifford modules:
The uniqueness of the spinor representation is a consequence of the fact that the algebra of matrices is simple, that is, it has a unique irreducible module.
I also looked into Nicolaescu's notes and he says that
the uniqueness of the spinor module follows from Schur’s Lemma
(see below for Schur's Lemma in the form considered by Nicolaescu.) These two explanations are not detailed enough for me, can someone elaborate please?
Schur's lemma. Let $E,F$ be simple $R$-modules. Every non-zero homormorphism of $E$ into $F$ is an isomorphism. The ring $\mathrm{End}_R(E)$ is a division ring.