What Blossom does is different from proving ownership. It simply proves that the contents of the referenced media has not been altered from its original form at the time it was uploaded to the Blossom server, and is therefore exactly what the author of the Nostr note intended others to see.

This is because the file name for the uploaded media is just the hash of its content. Therefore, when a Nostr client makes a request from a Blossom server for that file in order to display it to the user, it can easily check whether the hash of the content matches the file's name. If it does, the media has not been altered. If it does not match, then the client should not display the media and should show a warning, such as the red icon nostr:npub16jdfqgazrkapk0yrqm9rdxlnys7ck39c7zmdzxtxqlmmpxg04r0sd733sv mentioned previously.

Of course, this offers no proof that the video was not doctored, modified, or otherwise "fake" prior to the time it was uploaded.

So, you have no assurances that the person who uploaded the video is the "owner," or just reposting it from somewhere else, and you have no assurances that what the video shows is not fake. You only know that what the author of the note intended you to see at the time they created the note has not been altered, or else it would not have been displayed to you by your Nostr client.