I am struggling to find information on the Stacks project, so I am crowdsourcing here. (Sorry.)
Let $\mathcal{X}$ be a stack (algebraic, if it makes a difference) over a category $\mathcal{C}$ (e.g. $\operatorname{Sch}/S$) and let $U$ be an object of $\mathcal{C}$. There is a sheaf $\underline{\operatorname{Isom}}(x,y)$ associated to any two objects $x,y\in \mathcal{X}_U$, where $\mathcal{X}_U$ is the fibre category of $\mathcal{X}$ over $U$. This sheaf assigns to an object $V\overset{f}\to U$ in $\mathcal{C}/U$ the group $\operatorname{Isom}(f^*x,f^*y)$, which is the group of isomorphisms in $\mathcal{X}_V$ over $\operatorname{id}_V$. (The fact that it is a sheaf is part of the definition of being a stack. For algebraic stacks, it should be an algebraic space.)
I'm not sure if this is true, but I believe it to be true by unraveling some definitions: if $x,y\in \mathcal{X}(T)$, then $\underline{\operatorname{Isom}}(x,y)$ is the pullback of the diagonal $\Delta:\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ by the morphism $x\times y:T\times T^{\prime} \to \mathcal{X}\times\mathcal{X}$. If you specialize to the special case $x=y$ then you will get $\underline{\operatorname{Aut}}(x)$, which is the automorphism group algebraic space of the point $x$. Normally, you get this by pulling back the inertia stack by the point $x$, but it seems to me that this is giving the same thing, judging from how I unraveled the definitions.
Can anyone verify this?
(Also, I want to verify something else: $\mathcal{X}_U$ is different in general from the $U$ points of $\mathcal{X}$. The former are $U$ points, but the morphisms between them are isomorphisms which project to the identity of $U$. The latter has no restriction on morphisms.)
