Why Chrome asks to store some website passwords, but not for other websites?
Sometimes, this is because some sites uses redirects after the logging in. The page where you logged in dissapeared so quickly, so the "save password" dialogue didn't have the time to appear.
I tried a lot of options, from clearing browsing data, resetting Internet Explorer options, to importing passwords from Firefox.
Then I agreed to sync my passwords in Chrome (which was disabled earlier). Suddenly, Chrome started to ask me to save passwords again for this particular site.
Posting this to help out somebody who has had a hard time like I did.