Ok, this is what you are doing, simplified version.
You have a character named X. X is your main dude.
You also have character Y, Y is your alt. Y is just a peasant and you only logged in with Y once from ingame and thus a character named Y exists.
Since you never logged into the webpage with Y, he does not have a password.
When you so log in with X you get a message that you have a character (mr Y) that does not have a password. This means, if someone, let's call him Mr.Haxors, knows the name of Y, then he can login with the name of Y and choose a password for that character. He can then switch to your main (X) and sell all your stuff/retire/sell your heirlooms to him etc.
So, when the link brings you up to the login page again it wants YOU to login as Y so that Y gets a password and nobody else can set a password for him. But you continue loging in as X so Y still does not have a password.
There are two ways you can fix this.
A) Login as X, press settings and check the box for change password for all characters. Then write down the pass pluss confirmation (the password again).
B) You get the name of Y, go to the login page and login with him with a password of your choice.
B is the most simple, but A is the most secure can't-possibly-fail way.
Hope this helped.