Reporting on behalf of fa:User:Arfarshchi per this comment on my talk page.
The user has set their preferences to use Jalali (aka Persian) calendar and this correctly reflects in Watchlist, RecentChanges and page history. But notifications from Echo are still showing him dates in Gregorian (default) calendar.
Core and Echo both use moment, so https://github.com/jalaali/moment-jalaali may be helpful.
@jmatazzoni The Language team is not planning to work on Echo, but can provide advice. Please clearly state what you are looking from us.
My understanding is that the calendar is not really part of Notifications but is powered by a more fundamental layer of the system. The thinking of the team was that if WMF wants to allow for alternative calendars, then that goes beyond Collaboration Team's scope of authority and responsibility. @Catrope could explain better. I see this is already part of "MediaWiki Internationalization" and "Mediawiki User preferences." I don't know what team has responsibility for these project, but maybe this ball is properly in their court?
Potentially interesting tasks: https://phabricator.wikimedia.org/maniphest/?ids=155824,101442,99674,103302,64900,17311,10385#R
One quick thought is that you use Moment.js to format the dates. I am not sure whether it supports other calendar formats as we do in PHP side.