The Discord<>Spotify integration is specifically implemented on their side as I understand things & isn't a nice way for others to make use of it (e.g. not being able to set "listening to" instead of the hard-coded "playing to" for the overall status).
For other app integrations, there's not a formally defined way to do it when I last looked into it a year ago but there seemed to be some hacks that could possibly achieve it but it requires uploading things from the WACUP client to a server to then host those images & fiddle what WACUP sends on the client update.
Without looking into things again I don't know if anything has actively changed on the Discord api side of things to formalise what can be done or if it's how it was when I last looked.
-dro