One side of the island connects with the Pont de Bir-Hakeim Bridge (the one near the metro station by the Eiffel Tower). The other side of the island connects with a bridge where the Statue of Liberty is located.
Yes, we were not mistaken. Though the original Statue of Liberty is located in New York City, smaller replicas of the statue are located all around France. The most famous one is erected here, on Swan Island. The statue faces towards the Atlantic Ocean, looking at the original statue in New York City.
The island itself has green trees, and during summer and fall these will be a beautiful background for photos of the Seine River, as well as Eiffel Tower photos. To relax and enjoy the peace and quiet, you can sit on one of the benches on the island and stare at the Seine River and the boats sailing on it.
One of the island's advantages, is that its less visited and most people who come here are locals. It is a place to taste the authentic Parisian experience, the less touristy and less busy side.