What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get $\mathbb{R}P^2$. So I believe one description is $\mathbb{R}P^2$ with two points identified, which is homotopy equivalent to $\mathbb{R}P^2 \vee S^1$. Is this the best we can do, or is it something else in disguise?
This answer suggests that it should be a surface, but I'm having a hard time seeing how it is. There is no neighborhood of the glued point homeomorphic to $\mathbb{R}^2$, is there? If so, how?
I'm putting what I have wrote in the comments together in this answer :
Claim The polygon $abab$ with all the vertices identified is homotopy equivalent $\Bbb RP^2 \vee S^1$.
First consider the polygon $abab$ with diagonally opposite pair of vertices identified, which is homeomorphic to $\Bbb RP^2$. Since all of the vertices must be identified, attach a $1$-cell to the resulting space along the disjoint union of the unidentified vertices; nullhomotoping the attaching map along the surface of $\Bbb RP^2$ gives you the desired homotopy type.
You can see that this is not a surface by looking at the wedged point. Any neighborhood of the wedge point looks like an open disk wedge $(0, 1)$, which is certainly not homeomorphic to $\Bbb R^2$.
As for the polygon in the MO question, the standard convention for identification on the polygon of $abab$ is that you identify diagonally opposite pairs of vertices, so that's just the usual $\Bbb RP^2$.