I would imagine that this is actually held in config files held on the PC? As far as i know these files are not accessible through the PS4 - so it's really a console issue not an ED issue.
I think that this is a valid question and potentially something that would benefit PS4 console players.
I am basing my theory on the functionality provided by the EA Fifa games. They have a dedicated settings file known as 'Hospitality settings'. This is the profile configuration or things such as half length, camera angles, referee strictness etc. It is a profile held on the central EA servers and downloaded to any console that you log into to override the default settings.
The majority of people will never have needed / used these settings as they will stay on the same PS4 / Pro console however, if you create a profile on a friends PS4 and boot up the game from within that profile EA will download these settings automatically.
If FDev could implement something similar (if they haven't already got it on the plan) then it could be used to keep a track of controller bindings.
Worth asking the question.