From the point of view of the server, having a single user from different stations would be just like having the same user with more than usual processes. IOW: Nothing special.
OTOH, maybe there are some applications that depend on having a single instance. Firefox for example does try really hard to have a single instance per user; but I don't know if it's 'dangerous' or not to have more than one. Another example would be the desktop manager itself (Gnome, or KDE, or whatever). Although they should work, there might be some rough edges.
In both cases, it seems that what they would be 'protecting' from the concurrency are the temporary work files, and not any persistent data.
What I would do is just be sure that the
/tmp directory is local to the client, and not bother.