I found that video very compelling. If it was eddy current, the float standoff distance would have decayed. It sure looked to me like there was no decay at all, and wow! if true.
Eddy currents can be induced in non-superconducting materials that make it look like levitation, but the catch is that there has to be relative motion between the magnet and the object 'levitating' to generate the currents in the first place.
But in this sample video, the standoff distance doesn't appear to be slowly dropping at all, which would rule out eddy currents as a source of the behavior. If you continue watching the linked video to 13:27, he talks about how and why superconductors levitate.
