Hmm, at a guess, it's one of the following to blame:
1. We need an updated Sonic Engine
from Sonic Solutions (they haven't provided one).
2. The CD/DVD-RW drive is incompatible with the Sonic Engine
3. Some concurrently running process is interfering, either:
a) Some other app which manages CD drives or maybe even also uses the Sonic Engine (eg. Adaptec Easy CD Creator,
Sonic RecordNow or MyDVD, MS Digital Photo Suite, etc) - make sure these are closed before installing Winamp, including any system tray icons... or
b) Some security app is preventing the Sonic drivers from being installed.
4. Or more than likely, for some strange reason, the internal Sonic installer isn't being run in admin mode,
in which case, see below...
Other than that, Sonic installs fine on other peoples' Win7 machines,
so it can only be something system-specific causing it.
You could try manually extracting the Sonic installer from inside the Winamp installer, eg. by using 7Zip
(v9.09b), and then running it afterwards...
1. Open the Winamp installer in 7Zip
2. Double-click the  folder in the zip
3. Then extract all files in the $PLUGINSDIR subfolder to some new folder on your HD
4. In said folder, right-click pxsetup.exe and click "Run as administrator"
5. Hopefully, the Sonic engine will now install properly.
Note: If you click OK on the dialog in your screenshot, then Winamp will still install correctly, and everything should still work except for CD burning (ripping will still work if using Windows native SPTI instead, though you might need to uncheck 'Use Sonic' in CD Ripping settings first).