Honestly, I don't think this is really preventable. We could look at the criteria and do something with a special case, but I don't think that's something we want to get into; there are likely various things that you could conceivably come up with that show some amount of information that is otherwise privacy controlled. Worth noting that this won't reveal the year, just the month and date.
As such, I think this will be called a design issue. If it's significant for you, you can likely hide the date based on trophy ID, though even then, depending on what other trophies are awarded, the date may be implied.
I can see a few borderline uses for a birthday trophy, but due to the way trophies work, I'm not sure how much it makes sense compared to, say, a membership length trophy.