I have a signal sampled from $\mathbb{R}^2$ at a frequency of $2\text{Hz}$. It is quantized from $\mathbb{R}^2$ to $\mathbb{N}^2$, seemingly using the round operator.
How can I estimate the error variance from this setup ? I've found models for univariate signals but they don't factor in the sampling error, and I'm not sure how I can simply generalize the quantization error from $\mathbb{R}$ to $\mathbb{R}^2$.
Let's call your signal $\vec{x}$ and the quantized signal $\vec{x_q}$ . If you define the error simply as the mean square error between those, i.e.
$$E = \left\langle\left\lvert\vec{x_q}-\vec{x_{}}\right\rvert^2\right\rangle $$
the error will (in most cases) be simply twice the error for the one dimensional case. Assuming rounding and a quantization step of $\Delta$ the 1-D error would (in most cases) be
$$ E = \frac{1}{12} \Delta^2 $$
"In most cases" means the signals are significantly larger than the quantization steps, the two dimensions are uncorrelated and the probability density function of the original signal is reasonably continuous. In this case the quantization error is uniformly distributed on [$-\Delta/2,+\Delta/2]$.
If you want to calculate the magnitude or phase error, there is information about the signal required.
