Electronic device operating temperature declaration
The operating temperature range specification is generally determined by testing a component. You generate a list of critical characteristics for the component and then you measure those characteristics while the component is heated or cooled to the specified temperature. If the characteristics are within their specified limits then the component passes the test and you can be confident in using those temperature extremes in a published specification.

However, the manufacturer can make an economic and marketing decision on what temperature range they want to use in their specification. If your target market is video game controllers, why go to the considerable expense of testing your product at -40C, even if you think it will probably work just fine at that temperature?

Now, if you design a system using components with different specified temperature ranges then you can at least be confident that the system will work within the most conservative limits of any component. But it is possible that some of the components will actually work just fine, at least well enough for your system, at temperatures beyond what the component manufacturer specified. If you test your system at the desired temperature range and it meets all of your specifications then you can reasonable claim that it will work over that temperature range.

I should say that like all component testing you must test not just one unit but a reasonably sized sample and perform the proper statistical analysis.