For any section of road there is usually one name in the map database, although it might carry several different numbered routes and one or more street names. The folks who work on the map database try to present a name you are likely to see on signage locally, but mappers are still well short of being complete on ground truth.
I find this annoying when trying to follow routes on Interstate and numbered U.S. Highways, when the route showing up on the GPS is different from the one I am following. And of course, in most urban areas, when numbered routes follow local expressways, what gets presented are the expressway names, if that is what is most used (and what is on the signage).
So generally, there is not an option to change the names being used. You might suggest changes to the company supplying the maps, but don't expect them to automatically substitute your naming preferences for names generally used locally. Most sat-nav assisted travel is local, and in many urban areas people prefer names they use locally.
It will be interesting to see how Garmin databases change as they begin to incorporate map data from the DeLorme acquisition, as DeLorme used some different naming conventions.