So, I think this bug has been around for many years. Curious it was never fixed. I called support and reproduced it. Domain codes are exported when exporting data to a table or a Shapefile. Yes, even when the Environment setting to transfer Domain Descriptions is checked. This is true:
In ArcMap 10.8.2 and in Pro 3.3.2. From both SDE (SQL enterprise geodatabase) as well as from file GDB from both ArcMap and Pro. Anyone else having this issue? And I also tried copy and pasting from the attribute table out of both, into Excel. All of the above, fail. Only code is exported. Bug? Esri team, this is important and affects nearly every customer. Support had no workaround or solution.
(update: bug is still in 3.4.3 in May 2025...)