No, there is no coordinate system. This is referring to the distribution of a test statistic for hypothesis testing. It's a 1-d real scalar, and coordinate transforms don't have any meaningful statistical representation. Of course there are much higher-dimensional distributions, in all sorts of coordinate systems, involved in sampling the test statistic, but at the end of the day this is all you are left with. If you change the underlying distributions of the model, then of course you will change the test statistic distribution, but that's meaningless, since the whole point of the test statistic is to quantify an observation in the context of a given model.

Anyway, as I mentioned elsewhere, the motivation for calling it sigma is that, by construction, it maps onto the quantiles of the standard Normal distribution. So an N-sigma result will have the same p-value as N standard deviations in a Normal distribution. So you can associate "sigmas" with "standard deviations of the Normal distribution". Perhaps this is what you are trying to say, but it does not make sigma a standard deviation in any statistical sense, i.e. it is not necessarily related to the variance of the relevant distribution.