Hi Alexander,
Yes there are limitations when using folder based sub sites. I have updated the documentation here. I thought I had already documented this but apparently not.
The issue is that when using folder based sub sites as compared to host name based sub sites is that folder based sites share the same physical cookie file and the same authentication ticket.
I tried to find a solution to make it not show the name in the LoginName control but the idea I had didn't work. LoginName control is not used in mojoPortal, I don't know a solution to make it work as you would like in your custom code. Apparently LoginName just shows whatever value is in Context.User.Identity.Name without first checking if .IsAuthenticated is true. My only suggestion would be to in code set the .Visible property of that control to IsAuthenticated
Hope it helps,
Joe