Sometimes I can't save log-in user names and passwords for sites, even if I click Save when prompted. For example for Flickr. I guess it's an Epiphany bug, but I couldn't find any information about it.
You can track / report epiphany bugs on the GNOME bug tracker site:
I do not think this is a bug. Flicker needs a Yahoo account. Yahoo passwords are not stored (for security reasons, I think). First login to Yahoo and indicate that you will keep signed in. As long as you are logged in your Yahoo account, you will not need a separate login for Flicker.
BTW you can register or update your password in epiphany by entering your credentials on any site. The only missing function is that you can not view your saved passwords - or I do not know how to view them. So, for simplicity I regard this as a security feature. ;^)