Is there a name for categories $\mathcal C$ such that $\text{Obj}(\mathcal C)$ coinside with $\text{Mor}(\mathcal C)$?
In the diagram below the morphisms are $a\overset{a}{\to}a$, $a\overset{b}{\to}c$, $d\overset{c}{\to}c$, $a\overset{d}{\to}e$, $e\overset{e}{\to}f$, $c\overset{f}{\to}e$.
I don't know if this is interesting mathematically. My idea is modelling. In the models all objects should be of the arrow-type in the same category. Nothing in the definition of categories exclude this. Domains and codomains can be defined as usual.
Interpreted in a certain way what you are asking is standard. A category $C$ is a collection $C_1$ together with two maps $s,t:C_1\to C_1$ and a map $m$ from $C_1\times_{\langle s,t\rangle} C_1 =\{(f,g)\,|\,s(f)=t(g)\}$ to $C_1$ satisfying: $st=t$, $ts=s$, $m(m(f,g),h))=m(f,m(g,h))$ and $m(f,s(f))=f=m(t(f),f)$. Note that normally we write $m(f,g)=f\circ g$.
To obtain the usual description of category set $C_0=\{f\in C_1\,|\,s(f)=f\}$, keep the same composition, and let the domain and codomain maps $c,d :C_1\to C_0$ be defined by $c(f)=t(f)$ and $d(f)=s(f)$ respectively.
Conversely given a category $C$ forget the set $C_0$, keep the same composition, and let $s$ and $t$ be the maps defined by $t(f) = 1_{c(f)}$ and $s(f)=1_{d(f)}$.