The way it works is that IIS is simply a proxy to Tomcat, so you only install the extension on Tomcat – never on IIS. The IIS connector needs to know how to handle the WebSocket connection but that’s a different story.
IIS listens by default on port 80. Tomcat by default listens on 8080 or on some Lucee set ups on 8888. If you have Tomcat listening on port 80 then you’d have a problem because it will “fight” with IIS on the port, and the first one that takes the port “wins” while the other will fail to start, so there’s defeinitely something weird with your setup.
It makes sense to delete anything you don’t need and don’t use, but as always – be careful to only delete stuff that you don’t need. It’s always a good idea to make a backup before you delete anything in case you do delete something that you need by accident.