If you are not using the default install of Ubuntu, you might not be using Gnome. This is the case for flavours such as: Kubuntu (KDE Plsama), Lubuntu (LXQt), Xubuntu (XFCE), and so on.
And if you have installed Skype using
snap, then you will likely find that it will always ask you to login upon loading, even if you have it configured to automatically run at login.
The issue is that Skype installed via snap relies upon
gnome-keyring, but fails to install it as a dependency.
Thankfully, one can install this dep. without bringing in the entirety of Gnome.
Open a terminal, and enter:
sudo apt install gnome-keyring
Restart (or start) Skype and login.
The application should communicate with the
gnome-keyring service and securely store your login details.
Quit Skype and reload it. You should find that you are not asked for your login details again.