If the issue is with e-mail notifications, first of all make sure that you're checking the right e-mail inbox. It should be the same as the one in 'My email for notifications' field shown in the screenshot.
If that's the case, check your Spam folder for notifications from ManyChat - they could have gotten there by mistake.
What's for not receiving Messenger notifications, make sure that you have opted-in to them under Settings -> Notifications -> Messenger Notifications. If you're having troubles opting-in, check out this article.
After subscribing to Messenger notifications, be sure to check the box next to 'Messenger' under Settings -> Notifications.
Also, make sure that you're chosen as an admin who should receive notifications in the 'Notify Admins' block in a Flow.