Identifying two points on a surface

Question

seen an argument that if you have a path connected surface and you identify two points on them, you can use a CW structure to show that it is homotopy equivalent to the original space wedge a circle. I was wondering if the argument "generalises" to identifying pairs of points, specifically whether we can do this arbitrarily. My first thought would be you can do it finitely, possibly countably but not uncountably (take $\mathbb{R}P^2$ and its structure as the quotient on a sphere).