They use the sidestore:// and altstore:// URL schemes. If you don't have that app installed, iOS has nothing to hand the link to and shows an error.
Copy the URL above instead.
In SideStore or AltStore: Browse → Sources (top right) → + → paste.
Before you install
Ammy needs a relay running on a Windows PC. The app reads your now-playing track and sends it to a small Python program on your desktop, which is what talks to Discord — Discord has no Rich Presence SDK on iOS, so there is no version of this that works from the phone alone. Presence only appears while that PC is awake with Discord open.
Setup is genuinely involved: a Discord application of your own, Tailscale for the tunnel, and the relay installed as a startup task. Read the README before installing the app, or it will do nothing.
You need SideStore (or AltStore) already set up with your own Apple ID.
On a free Apple ID, sideloaded apps stop working after 7 days until refreshed, and count against a 3-app limit. That's an Apple restriction, not something this app can opt out of.
Manual install
Download the .ipa and add it through your sideloader's + button under My Apps.