Let me start by interpreting the question "What kind of surface is $S$?" in the case of a general connected oriented topological surface (without boundary). (I am considering only oriented surfaces just for simplicity of discussion.) If $S$ had finite complexity, i.e., would be homeomorphic to the interior of a compact oriented surface, you probably would be satisfied by the answer of the type "$S$ is has $n$ ends and genus $g$", since this provides a complete set of topological invariants. Surfaces of infinite complexity are also classified by a certain set of invariants:
Its set of ends (regarded as a topological space).
Its genus.
Its set of ends with positive genus.
You can find more details and references in this MO post.
If you look closely at the surface you are interested in, $H^2/[G,G]$, you realize that its invariants are:
The surface is 1-ended (simply because the abelian group $G/[G,G]$ is 1-ended).
It has infinite genus (this is easy to see and is explained in Sam's answer).
In particular, its only end has positive genus.
To summarize: Your surface is the unique connected oriented topological surface of infinite genus and one end. If you are looking for a different answer, you should clarify what does your question really mean.