I may be way off base here... but 'internship' and 'second project' seem like important clues here.
You finished your first internship project already, before the end of June. Well done. Classes start on August 18th, right? Most internships don't last right up until the day before classes start, so I would assume your last day is August 12th, or earlier. That gives you 6 more weeks, maximum.
On the other hand, each of your three points above suggest that there are multiple people in each department that need to be contacted, and at least 3 separate departments that need to be coordinated, prior to actually getting any work done. If you are involving at least 6 different cross functional resources in a matrix or functional environment, especially with physical resources used cross functionally, this is no 6 week timeline.
Your manager doesn't want you to talk to these people. Because you won't be the one whose trust they require down the road, and you certainly won't actually be acting on what they are saying. Your manager is worried that they would have to repeat themselves later (no one likes to repeat themselves), making the project that much more difficult to complete later on.
Your manager wasn't prepared to have an effective intern, had only one summer project, and now has absolutely no idea what to do with you.
Go ahead, be upset for a few minutes. I too really don't like not having anything useful to do. Then realize that you have a golden opportunity here. Well, first confirm with your mentor that this is what is going on. Then, realize that you now have over a month to pad your resume and build your network. Find things to do. Become a superstar, and get recommendations from people in multiple departments out of this internship. Explore the company as a whole, and learn what you do and don't like about the entire company, not just your little department where you have been for the last 6 weeks. Don't goof off, obviously, and do coordinate efforts with your mentor, but it is time to be self directed with your career growth, and to use your mentor's connections as leverage. You won't have this kind of time (40 hours/week) to freely network and do job exploration for a long time.
So, document well what you have already done with this project. Maybe put it in a fancy binder, or whatever will impress your manager. They want that, it will save them time come September when they will be pursuing this project in force. Keep in touch, ask periodically if there is anything you can do. But don't annoy them, because then you become both useless and annoying to them, the manager gives you busy work that won't look good on resume or networking, and then your adviser gets upset because they can't place an intern there next year.