Topologically the scheme $\rm{Spec}(\mathcal{O}_{X,x})$ is exactly the intersection of all neighbourhoods of $x$ and algebraically it contains every infinitesimal neighbourhood of $X$.
Although technically it is not the germ of$X$ at $x$, it seems to me that it contains so much information about that germ that it could be considered a materialization of that germ .
Also: technically it is not a subscheme of $X$ (since it is not locally closed) but I can imagine a world where a broader notion of subscheme would allow the monomorphism $\rm{Spec}(\mathcal{O}_{X,x})\hookrightarrow X$ to be called a subscheme.
In particular, given a morphism of rings $A\to B$ , the corresponding morphism of affine schemes $\phi:\rm{Spec} (B)\to \rm{Spec} (A)$ and a prime ideal $\mathfrak p\subset A$, the morphism $\rm{Spec} (B_{\mathfrak p}) \to \rm{Spec}(A_{\mathfrak p} )$ is a pleasant thickening of the genuine fiber $\phi^{-1}(\mathfrak p)=\rm{Spec}(B\otimes_A \kappa(\mathfrak p))$ of $\mathfrak p$.
Considerations of such almost germs $\rm{Spec}(A_{\mathfrak p} )$ of $\rm{Spec}(A)$ at $\mathfrak p$ may be of help when following reasonings in commutative algebra.
My first contact with one of these almost germs was in Mumford's Red Book, Chapter Two, §1, Example F, where he describes an example as "... a startling way to make a scheme out of the non closed points in the plane" and draws one of his celebrated pictures to illustrate the notion.