The doctor was curious to know if he indeed was a royal or not and if he really was then he wanted to know the reason for wearing the torn and tattered clothes.
But Shiro couldn't possible answer that since he was aware that it would grab too many attention. Also, he had no plans on revealing it anytime sooner since he was already deemed to be dead.
Everyone thought that he had died the moment he had entered the Dark Fire forest but he was perfectly fine. He basically exited the Dark Fire forest and then entered another deadly forest out if which also he exited.
So if he were to literally announce that he is the heir of the Dark Fire family then everyone will come running towards him and would start to ask millions of questions and at a such a time when he had to stop the Invadous, he c