Gmail Notify is small extension for Gnome 3 shell. It checks for new mail on accounts registered with Gnome Online Accounts. Because Ubuntu version of GOA supports only Gmail at the moment it's GMAIL notify. As GOA team will add other email / social networks providers extension will support them and will evolve into GOA Notify.

Gmail Notify is not solution for checking every type of email account and will not store any credentials. Email providers like Yahoo, Hotmail started to support OAuth authentication so Gmail notify will support them in the near future I think.