Edit: There has been some confusion with the construction below. I think My question can be summarized as:
"How do I characterize a manifold quotient whose group action in not finite?"
Original construction: I start with the hyperbolic disc $\mathbb{H}^2$ and partition the boundary into 4 sets of equal length $\{A_i\}=\{A,B,C,D\}$. Choosing to parameterize the boundary by an angular coordinate $\phi$ I have $A=(0,\frac{\pi}{2})$, $B=(\frac{\pi}{2},\pi)$... etc. Call the geodesic with endpoints on $\partial A_{i}$, $m(A_i)$. By symmetry each $m(A_i)$ has the same shape and length.
Now suppose I define the equivalence relation $\sim: \; m(A)\sim m(B) \sim m(C) \sim m(D)$ and define the quotient space $ Q \equiv \mathbb{H}^2 \setminus \sim$. This has the effect of creating 4 sheets, one for each region of the manifold between $A_i$ and $m(A_i)$ which I call $r(A_i)$. These meet at the common locus $m$. Notably this removes the interior region $\mathbb{H}^2 \setminus \cup_i r(A_i)$.
What is the correct way mathematically to describe $Q$? It's not a manifold, but each leaf can inherit the metric and coordinate charts from $\mathbb{H}^2$ except on $m$. Even so $m$ has a well defined length. In this (imprecise) sense $Q$ is "almost" a manifold. Is there a more general theory of such objects?
Edit: For reference the construction I'm interested in can be viewed as a generalization of pg.7 fig.1 of 1406.2663. The distinction being I am interested in the case when more than 2 geodesics of the same length are identified.