I do not think canceling the login session is the right thing to do. A login is independent of previous sessions and it should be so.
If you want to maintain some information about the logged in user as a profile, you could make it work in different ways. One that is prevalent in the industry is asking the user if the terminal they are using is their home/work computer. Thereby one can decide if they want to maintain some sort of profile information on that system.
If the user selects remember me, or that this is their personal device, the system may maintain some data on the system. If they decline this offer, then when the user logs out, all information of that user is removed from the terminal.
Important point is, different login sessions should be kept strictly independent from each other.
Also, it is not clear what are you storing as user profile and does the user have any visibility in what is stored? It is important that no user should see the profile of any other user, and if seeing his or her own, there should be an independent way of clearing his own profile as well.
Self-clearing can be optional as I think asking if you want to clear profile information on every login is going to adversely impact user experience.