The principle way is to make a statistical model for the error sources and than estimate the contributions to the errors. The estimates can be from component spec, measurements of sample units and other special test. This can also include measurements on each units, to check some relatively easy to measure parameters (e.g. initial offsets or bias currents).
Experiance with earlier models can also help to estimate those parameters.
Not everything can be reliably measured and is strict science. Especially long term drift estimates with newer parts are difficult and there is some room to get more or less stringent or confident in the performance.
The statistical model can than be used to calculate uncertainty limits to a certain failure properbility.
The final specs are than usually a simplified model, e.g. specifying the limits as part of the full scale and actual value to make them easier to use and more comparable.
Depending the company they may not do whole strict math (or do it similar with different naming or context - math sometimes has several ways to lead to the same result, giving different named to intemediate results). Worst can the specs are more from marketing and the dev. team only checks for obvious errors.
I just would like to know how it is calculated so I could in theory create my own specification by doing my own testing. For example, plenty of meters have a spec for +/-%5C with a spec for every 1C outside the spec. If working to a lab temp of +/-2C could I reduce the spec by a multiple of the 1C spec?
It does not work this way: The TC specs are upper limits for a quite large range. It is quite common that the TC is highest at the extremes of the temperature range. They may very well have a more detailed model (e.g. TC depending on temperature range, include higher order terms) for the TC in the calculation. So chances are they assume less effect from the TC in the specs for the +-5 C range. The official specs may not show all the details used in the calculation. There may be an internal tighter TC spec for the close range, that is just not published.
So subtracting the TC part, even if done in the RSS way would be usually too optimistic.