I don't mean when are two quotient spaces homeomorphic ($\cong$) to one another.
It's clear that if $X$ is a topological space, and $A$ and $B$ are subsets of $X$, that $A \cong B$ does not necessarily imply $X/A \cong X/B$. For example, if $X = I$ is the unit interval, $A = I$ and $B = [\frac 1 2, 1]$, then $I/I$ is a one-point space whereas $I/[\frac 1 2, 1]$ is homeomorphic to $I$.
I mean instead that if $X$ is a topological space, and $A$ and $B$ are subsets of $X$, and $X/A \cong X/B$, are there conditions on what must be true of $A$ and $B$?
Let $X = I$ and consider $A$ a point and $B$ a small interval; this is very similar to your example.
For a positive result along these lines, look up the long exact sequence for relative homology.