Confused about the definition of a group as a groupoid with one object.
The single object in the groupoid corresponding to a group $G$ does not really correspond to anything in the group - but you can think of it as a thing which has as its group of symmetries the group $G$.
As an example, the category whose only object is the set $[n]=\{1,\dotsc,n\}$, and morphisms are bijections from this object onto itself. Then the symmetries of $[n]$ are given by the group $S_n$.
For a more subtle example, consider a topological space $X$, and fix a point $x\in X$. Consider the category with one object, namely the point $x$. A morphism from $x\to x$ is a homotopy-class of paths from $x$ to $x$. Composition of morphisms comes from concatenation of paths. The automorphism group of $x$ is the fundamental group $\pi_1(X,x)$, which is a group of symmetries of the based topological space $(X,x)$.
However, if you consider groupoids with many objects all of which are isomorphic to each other, then the different objects correspond to different realizations of the same group. Moreover, each isomorphism between different objects gives rise to an isomorphism between the corresponding realizations of the group.
To make this precise, suppose $\mathcal G$ is a groupoid where all objects are isomorphic. For each object $x$ of $\mathcal G$, let $G_x$ denote the group $\mathrm{Mor}_\mathcal G(x, x)$. A morphism $\phi:x\to y$ defines an isomorphism $G_y\to G_x$ by taking $g\mapsto \phi\circ g\circ \phi^{-1}$.
A nice example of this is the category $\mathrm{FB}_n$ whose objects are finite sets of cardinality $n$ and morphisms are bijections. Then the automorphism group of each object is isomorphic to $S_n$. Isomorphisms between objects determine isomorphisms of their automorphism groups.
Another example is the Fundamental Groupoid of a path connected topological space $X$. Its objects are the points of $X$. The set $\mathrm{Mor}(x,y)$ is the set of homotopy classes of paths from $x$ to $y$. In this groupoid, $G_x$ is the fundamental group $\pi_1(X,x)$ of $X$ based at the point $x$. Different base points result in isomorphic fundamental groups, and isomorphisms are determined by homotopy classes of paths between these points.
Disappointing as the answer may be, it doesn't really correspond to anything.
All the group axioms refer to elements of the group; and these elements correspond with the morphisms in the category. We don't care what the object is: there's only one object, after all! So all we're bothered about is the morphisms.
I think of a group as an abstraction of the invertible transformations of something. Since it is an abstraction, we don't worry about what that "something" is.
A one-object groupoid says exactly this, viz., that we are dealing with invertible transformations of "something". The single object of the groupoid is the "something".