This thread came up in my search for the same. The feature appears to have been removed from MS Edge at some point, but there is still a supported workaround through the registry.
An example for adding Gmail and Google Calendar is below. This is tested and working in Edge 143.0.3650.66. Navigating to edge://policy will show the RegisteredProtocolHandlers property properly applied with a status of "OK" on success.
The usual script and registry disclaimers apply: take backups, use caution, and make sure that you understand what you are running.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Edge\Recommended]
"RegisteredProtocolHandlers"="[{\"url\": \"https://mail.google.com/mail/?extsrc=mailto&url=%s\", \"protocol\": \"mailto\", \"default\": true}, {\"url\": \"https://www.google.com/calendar/render?cid=%s\", \"protocol\": \"webcal\", \"default\": true}]"
See Richard Courtenay's answer and my response here for more context:
https://learn.microsoft.com/en-us/answers/questions/2339128/where-do-i-find-the-protocol-handlers-in-edge?page=1&orderby=helpful&comment=answer-12280319&translated=false