Consider the unit interval $[0,1]$. If I glue the end points together through the usual equivalence relation:
for $x,y\in [0,1]$, $x\sim y$ $\iff$ $x=y$ or $\{$ $x,y$ $\}$ $=$ $\{$ 0,1 $\}$.
Then the quotient space $[0,1]\backslash \sim$ is the $"same"$ as the unit circle in the plane. Because end points are mapped to endpoints, does that informally imply that the unit circle is homeomorphic to a triangle, a square, a rectangle and etc?
Geometric shape is not a topological property. In fact, a circle, a triangle, a square etc. are homeomorphic spaces. You can write down explicit homeomorphisms if you want.
It does not make sense to associate a specific shape of a plane object to $[0,1]/\sim$.
The choice is completely arbitrary.