There are some instances I have noticed that you will not get asked to save a password. I don't think it is an IE9 issue though but a website issue itself. Haven't really looked into it to be 100% verified but I'm pretty sure. As stated by Nithin, Japp, in most cases you will be presented with a popup at the bottom of the IE9 window asking if you would like to save a password.
Also check in Internet Options\Content\AutoComplete that Usernames and Passwords on Forms and Ask before saving passwords is checked.
@Bellaar: While not everyone saves their passwords, there are users that do. The post was a question on How To save passwords and not Should he save passwords. Lets try to keep the forum questions to answers only as they are used for future reference to help other users as well.