Here is how to add a Shairport Sync device to the Apple Home application on iOS 15.3.1. Note that Shairport Sync has to be built with AirPlay 2 support.
In the Home app > Home settings "Allow Speaker & TV access", select the option "Anyone On the Same Network". If a "require password" option below the "anyone on same network" is available, make sure it is unchecked.
Tap the Add Accessory
panel or choose "Add Accessory" from the +
symbol in the top right:
Step 2: Tap More options...
Your Shairport Sync device should appear here, though you may have to scroll. Tap the appropriate icon and follow the remaining instructions.