detection of periodicities in n-dimensional signals
Would these be limited to detecting periodicities that are aligned with one of the basis vectors of the n-D space?


Well, if you say "sampling is sufficient", then it follows that the whole space is the span of the base vectors (in fact, you usually have them orthogonal).

From that follows that a non-base-vector aligned oscillation can be represented by a linear combination of oscillations that are aligned with base vectors.

From that follows that yes, with your requirement that sampling frequency is sufficient to begin with, the usual methods of 1D frequency analysis work.

Note that you can't generally detect all periodicities with just a sufficiently fast sampling – if something is periodic with a period  longer than you observation window, you have no information about that. So, observation length must be sufficient, too, unless your signal model restricts both highest and lowest frequency.