Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.

What is the name and topological structure of the generalization in which $n$ pairs of distinct points of a sphere are identified? Is the resulting topology independent of the relationship between the identified pairs of points? In other words, are all surfaces obtained by identifying $n$ pairs of distinct points homeomorphic to one another?

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I don't know that these objects have names, but identifying $n$ pairs of points on $S^2$ corresponds to taking a surface of genus $n$ and pinching off each handle. Equivalently, take a genus $n$ handlebody and collapse a compression disk bounded by the belt sphere in each handle down to a point, then take the boundary of the resulting object.
To see that all such surfaces (of fixed $n$) are homeomorphic, it suffices to see that for any two finite set of $n$ distinct pairs of distinct points $\{p_1,p_1',p_2,p_2',\ldots,p_n,p_n'\}\subset S^2$ and $\{q_1,q_1',\ldots,q_n,q_n'\}\subset S^2$, there is a homeomorphism $S^2\to S^2$ mapping $p_i\mapsto q_i,\ p_i'\mapsto q_i'$. Such a homeomorphism will descend to a homeomorphism of quotients. This can be done by composing a sequence of Dehn twists along narrow annuli, each containing $p_i^{(')}$, $q_i^{(')}$, and none of the other points to be identified.
Spaces similar to this arise in the study of Riemann surfaces: the only way a sequence of surfaces in the moduli space of genus $n$ surfaces can leave all compact sets is by this sort of handle pinching, equivalently regarded as the length of a closed geodesic tending to $0$. So you might find more information about these pinched surfaces by consulting a treatise on the ideal boundary of moduli space or of Teichmuller space -- I don't know any off the top of my head that treat this in particular, but I'd start by looking in Hubbard's book.