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Without knowing anything else about the dog, I would guess the dog is fearful and nervous, likely due to the whining and anxiety (real or perceived) exhibited by the crated dog, and is seeking comfort by going back and forth between the vet staff and the owner. The combination of the tucked tail and the continuous approaches make me think the subject dog is possibly looking to flee but doesn’t have a way out.
The scents of the vet’s office could also be contributing to the anxiety levels of both dogs. I think there are some places like vet offices, shelters, pounds, kennels, etc. that can overload some dogs with unfamiliar and/or unpleasant scents that cause a fear/anxiety reaction.
The constant movement may be the dog trying to find some physical security by going back and forth between the owner and the other human who is offering attention (protection).