By default, all dnn modules have a distinct user base, check out www.snowcovered.com for 3rd party modules, that allows sharing it.

You should be able to implement this but it may be difficult to maintain based on the number of users you may have.  Currently where I am working we have multiple sites with a few that are parents and some that are childern.  If l log into the parent at http://www.parent.com and then navigate to the child at http://www.parent.com/child DNN retains my login status and the child site shows that I am already logged in.  I think that takes care of the login memory function that you want.  We are currently running 4.4.1 so I don't know if this works in the most current versions but I think it still does.  The only catch is that you have to specify roles for each user on each site (portal) seperately.  I have a different role for the parent site then I do for the child site.  This can make it fairly difficult to maintain a large number of sites with a large number of users.  If each portal has a different set of roles then the users from Site A wouldn't be able to do anything on Site B and vice versa.  There may be a better way of handling this, and somebody from the community may have that knowledge, but it sounds like DNN will do exactly what you want from the get go.  The only catch is that it will involve some maintainence to keep up the users and their roles.