If we go back to the days when we had a tax disc on our windscreens it clearly showed the car was disabled exempt for any passer-by to see so you could say its public.
While nosey neighbours might put your reg into the checker to see it, if their lives are that boring, I wouldn’t worry about it myself.
The only thing that worries me when it comes to DVLA and the reg of our cars or anybody else is them selling our information to any private car park owner who use ANPR and hence the possible risk of abuse by people who are not in authority like the police.