You can be made in charge of a project all on your own so at the end of the internship, people will recognize that the project is yours. If you choose, you can work on existing projects with members of the team as well.
At Uber, as with any company, the amount of ownership you get over your project will vary. It's dependent on the team you get, the manager you get, and the project you get. For me, I was lucky enough to participate in the actual designing of my project, and I owned it from start to finish, even onboarding a customer at the end. It was an excellent experience and I know that it's definitely not something that every intern gets to do, so definitely keep your expectations in check. In general, interns can expect to have pretty solid ownership over the implementation and specific algorithms used. However, usually, the high level design will already be done by your manager.