This 2012 article seems to be a good overview:

https://www.caranddriver.com/features/a15117726/faking-it-en...

At that time, Porsche, VW, BMW, Ford, and Lexus had various types of system to enhance engine noise.

It seems Porsche and Ford channeled the real sound through mechanically. VW started off doing that but then switched to an electronic system called Soundaktor. Lexus hired Yamaha to do an electronic system. BMW did it electronically.

Things may have changed since then. The article makes it sound like Lexus did it before BMW. And probably VW too, although that's less clear.