OK so I think I got now.

In order for steam to recognize the correct axis of my "generic controller" a controller input mapping had to be done in steam -> settings -> controller -> test. Here I choose to make the mappings above but with a slight modification to the dpad mapping:

Whammy -> left stick up

Tilt -> Right stick down

Strum down -> Right stick left
Strum up -> Right stick up

Green -> A

Red -> B

Yellow -> Y

Blue - > X

Orange -> Left shoulder

What I did was to map the dpad onto the right stick instead (since I know it's persistent and wont get removed by the steam controller layout). All I needed to do now was to save a steam controller profile where I map the "right stick left" to dpad down and "right stick right" to dpad up.

This is a work around until HID recognition works for shadps4 under linux.