My recent experience with two projects with the same short project name wasn't a good one! Although it is possible to register the second project with an identical short project name, if a user is wanting to S/R both projects, it doesn't work. The only solution is to change the short project name of one of the projects - and this is possible even after registering / working on a project.
I believe that when S/R is used for one of the identically named projects, on the local machine the data is synchronised in the folder with that short name. When S/R is used for the other project, then that same folder is used. I'm guessing that either files are overwritten or else, if a different set of files exist in the two projects, then files from both projects might be contained within the folder.
I presume the data for the two projects on the server should be separate as each registration has a unique ID.
For your situation, I don't understand how the team can open Project 1 but not see the folder on the local machine. If they make a change (add a space, delete the space) and then save, what happens? Can they find a changed file on the local machine?