While I don't think the URL shortening matters too much, there's still another important attribute of using these shortlinks that manifests itself on youtube: They also omit irrelevant metadata like URL parameters that were specific to your session or the way you arrived at the site.
For example, youtube will often include extra garbage parameters for stuff like playlists - which the youtu.be shorturls omit, making them perfect for sharing videos.
These parameters are not apparent to those with whom you share the shortlink, but I'd wager that the site stores a mapping from the shortlink to the session in which it was generated. Which may be OK for you; I don't know.
Is the identifier unique? Does it have to be? Without switching to a different browser on a different IP addr and finding the same video again through a different search, would you even know?
Yes, the format is youtu.be/<video id> where <video id> matches the one in the long URL (youtube.com/watch?v=<video id>&extra patameters...) and is unique to the video.
For example, youtube will often include extra garbage parameters for stuff like playlists - which the youtu.be shorturls omit, making them perfect for sharing videos.