This feature was implemented in T56130: Provide web notifications when user is sent an email via Special:Emailuser
but the configuration was set to "disabled" by default. (E.g. mw:Special:DisplayNotificationsConfiguration)
(I forget why, but I suspect it might have been due to my bad advice/hesitation at the time?)
I suggest enabling the web-notification by default on all wikis.
Hewiki already changed it: T114982: Enable by default "Email from other user" notification on hewiki