These things that look like collars around the swans neck are actually bands.
When they get banded a US Fish and Wildlife Service band is usually placed around their leg, just the same way that other birds are banded.
The collar is just an extra band so that they can be identified by by sight or through a scope, something that would be very difficult on a leg band.
The color of the band identifies what type of swan that it is. Trumpeters have collars that are red, yellow or green. While tundra swans are given collars that are grey, blue or black.
Even though the collars are made of light plastic and are placed loosely around the neck there is some concern that the collars may present a safety risk to the swan. This is especially true in northern regions during the winter time. People have reported seeing swans with ice built up around the collar. The extra weight can affect their flight or worse yet can cause the swans to drown when it can no longer hold its head up out of the water due to the weight.