on link hardware/app?
or after connecting to host pc?

or in games?

steam link should give ps prompts on link hardware/app before connecting to host
and give ps prompts on host in bpm

but most games only show xbox prompts

if it is not seen correctly by link hardware/app
please give the usb vid/pid of the controller, and a link to buy it

You're right - wrong prompts appear in the Steam Link android app in the first place. When I go to Settings -> Test Controller, all buttons are working and are mapped correctly, but the prompts are for the Xbox controller. Hope these photos will help:
https://imgur.com/a/XfRslym

Also, here's additional requested info.
Vendor ID: 0x358a
Product ID: 0x0304
Link to buy: https://playbackbone.com/ca/products/backbone-one-usb-c-playstation/

I have the exact same issue and it's disappointing that almost two years later it hasn't been fixed.

I think the fact that the Backbone One changed from PID 0x0104 to 0x0304 with the newer revisions probably just got overlooked.

I've gone ahead and made the change upstream in SDL3 (the open-source library Steam Link's controller support comes from), so future Steam Link builds should recognize the second-gen Backbone One as well as the first-gen one.