The definition reads that "A Gextension of a fusion category D is a Ggraded fusion category C whose trivial component is equivalent to D." It seems like a priori there can be multiple Gextensions for the same fusion category D. Is that really the case (i.e. no reduction mechanism)? But there seems to be a "canonical one" at least, which has the same category D sitting on top of each component of the grading. Does this particular Gextension have a name?

A complete reference for Gextension of fusion categories is http://arxiv.org/abs/0909.3140 (see also http://arxiv.org/abs/0911.0881). The kind of examples that you are saying are $\mathcal D\boxtimes \text{Vec}_G$, the Deligne product of $\mathcal D$ with Vec$_G$. 

