The correct answer was already given by Jed Brown in a comment, but a brief explanation might help. Shape regularity enters into finite element methods when stitching together local interpolation errors to get a global interpolation error (which is an upper bound for the approximation error and hence, by Céa's lemma, for the discretization error). Basically, you get local interpolation errors by estimating the interpolation error on a reference element (using the Bramble-Hilbert lemma) and then transforming it to each local element. This transformation gives you
- the required power of the element size $h$ and
- a power of the condition number of the Jacobian of the transformation
on the right hand side of the estimate. In order to obtain a global error estimate with only this power of (the maximal occuring) $h$ on the right hand side, you thus need a uniform bound for this condition number.
For simplicial meshes, you can show (independently of the dimension!) that this condition number can be estimated by the ratio of the diameters of the inscribed and the circumscribed (or minimal containing) ball. (In two dimensions, this ratio can in turn be estimated by $2/\sin(\vartheta)$, where $\vartheta$ is the smallest angle of the triangle.) If this ratio is uniformly bounded for all elements, the mesh is called shape regular. Hence in this case the usual definition is already dimension independent. You can find a detailed treatment in Chapter 1.5 of Ern and Guermond, Theory and Practice of Finite Elements.
(For rectangular meshes, you would probably use bounds of the condition number in terms of ratios of maximal and minimal edge lengths, which also makes sense for arbitrary dimensions, although I don't have any reference for that.)
Note that this is a separate issue from mesh quality, which also takes into account the conditioning of the linear system arising from the discretization, and ignores the magnitude of the constants in the mentioned estimates.