It is a common reoccurrence in engineering because engineering applications deal with real world processes that require viewing variables as random fluctuations. This is especially true of random noise, and also engineering tolerances, (as you mentioned) that have a “statistical spread”. This statistical spread can be usually generalised to be quantified as “variance”. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. So you have electrical noise power which is actually power generated by a random electromagnetic process, then you have variance of component or dimensional tolerances that is also the cause of random component or dimensional imperfections due to random manufacturing imperfections, all varying with a particular “spread” from the average expected value.
So yes, it may be true that you can usually add noise powers as the sum V ² or as I ² (with Watts as a dimension) but they are actually just “variances” of a stochastic process with a particular spread (Gaussian or otherwise), and generally variances of whatever (noise, component tolerances, or other types of random phenomena).
Electrical noise is usually measured with an instrument that directly measures noise power (a spectrum analyser) in a given bandwidth, and from this you can take the square root to get the noise voltage or current, just as you would sum “variances” together and then take the square root to get the “deviation” of that random variable etc - all this providing the particular combined random fluctuations are not correlated! This is a first order assumption/approximation that is often done in Engineering to simply things but still retain some accuracy. If the random noises you are squaring, summing and then square-rooting happen to be corelated, then your above sum of square will not strictly hold true unless the correlation is small.
This would be the same for component tolerances etc. For the later, it is more common for component tolerances to be considered in worst-case tolerance analysis, in a traditional type of tolerance stack-up calculation. The individual variables are placed at their tolerance limits in order to make the measurement as large or as small as possible. Summing of variances and square rooting the sum makes sense if you know the tolerance “spread” (the variance or the standard deviation). Generally, components are specified with some known min and max values as tolerances.
Then on another note, there’s also the Pythagoras theorem c ² = a ² + b ² that kind of looks the same but is not a result of the root of the sum of the variance of random variables a and b. It’s got nothing to do with statistics and random stuff. Pythagoras was lucky that a and b happen to be uncorrelated ;-)