I’m trying to add a Matter device, and have Matter in my HA installation. When I go to add it, from the app, it tells me to download the app.
Your device on which you run companion app is too old and does not support Matter.
You can add a bluetooth adapter to your HA server and dedicate it to the Matter server, which can then be used to add Matter devices, if you do not choose to get a newer device for the companion app.