This prevents the operator from simply saying someone has won and never paying out. In theory, you could go around to everyone on the list and ask them if they won a lottery prize. If anyone denies it, the operator might be pocketing prize money.
It is clear from the available evidence that publishing the amount that someone has won does harm to the winners. It becomes feedstock for a "people to rob or defraud" list. The same principle is applied to the secret ballot, wherein anyone can find out if you voted in an election or not, but they cannot know whom you voted for unless you tell them yourself. You can, of course, lie to them if you desire.
Q: Ma'am, did you win the lottery this week?
A: Yes. My name is on the list, isn't it?
Q: How much did you win?
A: A dollar.
Q: You didn't win the jackpot?
That's a new car, isn't it?
A: Of course not.
I got a good deal on the financing, that's all.
Q: And the 12-foot bronze statue of
David Hasselhoff's Baywatch character?
A: I found it at a dump.
Q: Clearly, you also hired Peter Dinklage
to dress up as Tyrion for your birthday party.
A: That's just an enthusiastic cosplayer.
It's also difficult to enjoy being rich without making it glaringly obvious that you have money, so I'm not seeing any pressing need for the judge to anonymize this particular winner after the fact. It might be warranted for the state to look into legislation that places greater weight on the financial privacy of future winners.