http://support.diamond-air.at/filead...15e-A33-r1.pdf
If your setup was approved, you would have a supplement similar to the one above for a DA40 series airframe.
As far as i'm aware in the US, the equipment gets certified and that certification is valid to all aircraft the equipment is installed in.
I believe that even for a part 91 operator in the USA you would need an LOA for certain RNP operations.
In Europe however every aircraft+equipment needs to be certified.
And the crew may require training. We had to in order to obtain P-RNAV approval.
Can it ever allow you to fly (lateral aspect) of a VOR or ADF approach?
It can if the equipment is approved.