Exporting passwords

The application you are seeing is "Seahorse", which manages the gnome keyring. Its distributor's website tells us:

File Locations Gnome Keyring looks for passwords and encryption keys in the following locations: ~/.gnome2/keyrings: Password keyrings, keys and certificates.

The export option is only available for certificates, not for passwords.

To sum it up: Backup ~/.gnome2/keyrings, and you have your passwords.

In new versions of GNOME 3, the file is saved in ~/.local/keyrings or ~/.local/share/keyrings.

As of the date of this post, I'm using GNOME 3.28.2, and this is located in ~/.local/share/keyrings