You're question doesn't make an awful lot of sense (though maybe because as you said, you've paraphrased it).
Session Storage (presumably Session State), Application Storage (presumably Application State) and Profile Info (part of the membership/role provider infrastructure) aren't Azure specific technologies. They're part of ASP.net so this would also apply to a asp.net web app outside of azure as well.
For each of those things, there are different ways that they can be backed (in process memory, table storage, sql database, app fabric cache etc...)
I think the key words are "Insecure" and "All Users"
If it's for all users, its something that needs to be stored at an application level. If it's "Insecure" or data which doesn't contain any data which needs encryption then you have a few options but I think what they're getting at in the question would be an application profile info.
You can use an Application Level profile to store default values for all users of your website (things like default look & feel, default options etc...)
Have a look at this link here - How to assign Profile values?