For an invitation to automatically be added to a person’s calendar they have to set up their email account to automatically accept/add invites to a calendar (depends on the email service).
That may be a big ask for some folks who want to manage their calendar.
The suggestion of setting up a shared calendar also works since the additions would just sync across. However, this assumes that you’re both using the same email provider, or that you’re able to share a calendar between email providers.
Notifications would then appear at whatever reminder time is set in the event.
Notifications are a mixed bag. iOS Calendar only provides notifications for invites from iCloud and Exchange accounts. They rest have to be processed manually through email.
And that’s assuming that the person receiving the invitations have notifications enabled for Calendar.
Back to processing the invitation manually… Click on the invite in the email. This launches Calendar. Then accept/decline the invitation.
Sorry for being so lengthy, just trying to be comprehensive.