So on the steam versions of these games, if you use a playstation controller, the game recognizes that and gives you the playstation UI.

That isn't the case with GOG, I believe some of the games gives me an incorrectly mapped d-input UI (numbers instead of buttons). 7 gives me the xbox UI, that seems to work. This is when I have steam closed too.

Launching 3 from steam breaks the controls. 7 works like before, just with the xbox UI.

I would like it if these games had the same controller functions as the steam versions,

If you have a playstation controller and can't get it to work. Install DS4 Windows.. Never had a problem with it. It will not give you the Playstation UI but at least your controller will be recognized as an XBOX controller.

Yakuza does support DualShock (see PCGW) , with button prompt
DS4Windows is recommended, the first launch really help
But It's not required

You can change the output controller to either xbox or playstation, in DS4Windows
Which is nice if you want to use dualshock even for unsupported games

Make sure you don't have leftover from software that manage controllers
Its a pain to discover some soft hide your device from windows (like HidHide, HidGuardian, etc)

EDIT : I'm pretty sure there's no difference regarding the controller
I just launched the game without DS4Windows ; now I have a map so better by default imo
The difference is SteamInput, can it interfere?

EDIT2 : Avoid having steam open. Just connect your controller after closing it.

edit3: Well, using a dual shock 4 controller on my end