What geographic reference system (WGS84, EGM96, EGM2008 etc.) do airport towers use when measuring things like aiport elevation and current QNH for airport? Is there even a commonly accepted one worldwide? I couldn't find an explanation on this subject online.
I've only done passing research on the matter before, so if anyone knows more please chime in. As far as I know the FAA uses NSRS (National Space Reference System), and the NSRS currently uses NOAA GEOID models, although this is expected to be changed in 2022. I'm not sure which regulation covers airport surveys, but you can reference AC 150/5300-18 (pdf). ICAO Annex 14 doesn't seem to mention a specific vertical datum for MSL, but specifies that, independent of the method, surveyed altitude for precision approach runways has to be accurate to "one-quarter metre or foot".
Interestingly, some airports and navaids aren't even using the WGS-84 lat/long coordinates, much less a recent international geoid model for MSL. This is increasingly rare, but you can check a list of waypoints excluded by Garmin for non-WGS-84 compliance here and see a list of overall compliance by country here.
While it's always nice to have higher resolution MSL altitude models, we have to keep in mind their limited usefulness. Since airport's charted elevation is the reference point for barometric altitude in aviation, the only issues with an inaccurate airfield elevation are seen when comparing to elevation measured elsewhere. As a result airplanes a few miles from the airport may indicate less accurate elevation relative to the airport and local sea level (but would indicate accurately relative to other altimeters at the same location). An inaccurate survey would also affect GPS's geometric or "GLS" altitude, but that's basically only used as a cross-check.
I'd guess that in practice temperature and pressure variations over an area are greater than the variations in geoid elevation error over the area, and regulations already account for those inaccuracies, for example by requiring higher MDA when using an altimeter setting from a different airport. Additionally, some of the tools for measuring altitude on an aircraft have much lower accuracy than the <1 m accuracy of these geoid models and surveys. Even an ideal radio altimeter can only measure 1.5 ft accuracy and then only when close to the ground (according to Arinc 707-7), and un-augmented GPS and barometric altitudes are frequently off by several dozen feet. Aviation is designed with imprecise altitude measurements so sub-foot precision isn't necessary or practical away from an airfield.