Has she considered changing the name on the tag, just at work? Just put "Jane" or something generic, or "Sarah" so it sounds similar. But yeah, it unfair that she has to be the one to change.
I almost never give my correct name for informal use (I am not a waiter, but for coffee orders and the like). Gives me a marginal amount of privacy (I mean, who cares) and makes it less likely my name will be mispronounced/misspelt with my coffee order.
If I were in a service job with a name tag I would definitely not use my real name.
I have a name that is easy to pronounce but hard to spell correctly, so it's easier to give a more common name, than making people struggle and explain how to spell it.
At one camp where I was a counselor, all the staff had made up names. I guess it was so we wouldn't get tired of hearing our real names. Having worked at several camps as a teen/young adult, I wish all had done this.
If I were in a service job with a name tag I would definitely not use my real name.