I've seen deer play that way with each other. I have no doubt the young deer was joining in to play. It was great how the dog settled down at the first word from the owner so he wouldn't nip. Of course, deer don't play by biting like dogs do. The only similar play signals that appeared to be mutual with the dog and the deer were the flailing with the legs. The dog did less of that, but still...I think they understood each other and played well together. If the deer had been afraid, you wouldn't think he'd keep coming back for more. He just didn't look afraid to me.
Anyhow, that was an amazing video...very cute. Thanks for sharing.