Yes, Tonido will be logged in until you close the browser. If you have multiple tabs opened and tonido UI was in one of them, closing just the tab doesn't count as a browser close. You need to close the entire window. If you want to logout from the UI, click on the menu item Tonido -> Logout. This will close your session irrespective of whether the browser window/tab is opened or closed.

I typically run tonido UI on a separate browser window (no tabs) and close it when I am done using it.

I beg to differ that logout alone closes the session. For the main account details, possibly yes (I haven't really tried it much), but for the guest user, no. I can logout from the guest login but still access a shared file using the direct path URL until all instances of the browser have been closed.

All instances of the browser have to be closed for the session to be closed. I opened an IE8 window, Guest logged in, logged out, left that window open (but logged out), opened another window of IE, closed the first window and I was still able to enter the URL of a shared file and still be able to access it. All windows must be closed.

We use HTTP Basic Authentication for guest login which means that the session is remembered by the browser till you close the browser completely. This is not specific to Tonido.