I don't know, but I assumed Apprentice Bard was the name for the internal testing version, and that the name for the released version will just be Bard.
In other words, "apprentice" is just a cutesy variation on "beta", "testing", "pre-release", etc.
Here's my take.
"Apprentice is Barred from this AI race" is the phrase.
"You (OpenAI) Apprentice are Barred from this AI race, because, well you are just an apprentice"
To counter Sun Microsystems' dominance in the Java market, IBM named its IDE as "Eclipse."( blocking Sun).
PS: To be clear, that is not my personal opinion of OpenAI, I believe OpenAI brought so much innovation to this LLM space.
Just sharing what came into my head, why Google picked that name.