It's really only a thing very early on in the game, while you're still at the hospital. By the time you leave you're mostly past it and fully invested in trying to reconnect with your family. There are even multiple points where you basically ask yourself why you ever considered the idea in the first place.
It's relatively easy to just dismiss it as being a symptom of his total confusion due to the amnesia and not a huge part of the story overall.
something in a story doesn't need to be realistic, otherwise you can't have fantasy and science fiction. But it does need to be believable, or at least reasonably believable.
The usual words that come up in this context are verisimilitude or relateability.
A story doesn't need to be 100% accurate, but it needs to be internally consistent or it takes you out of the story. And you have to be able to relate to it on
some level or you'll have a hard time caring. Which is why most alien races in sci-fi tend to act exactly like humans and most fantasy stories ignore 90% of the radical changes that magic would cause in any actual world. Because changing too much too far away from what is "real" makes it hard to relate to.
It's also why most settings with fantasy/sci-fi elements tend to change only a few things but keep the rest of the world more or less exactly the same. Because changing everything makes the world
too unrecognizable, and the viewer stops caring about it.
On the flip side, at least
some suspension of disbelief is necessary, otherwise you'd never be able to enjoy any entertainment media again. Just as an example, almost no story involving amnesia
ever presents the amnesia correctly based on how it actually works in the real world. Because real amnesia wouldn't make for good stories. So we all sort of tacitly agree to pretend that the
trope of amnesia as presented in tons of movies, shows, and books is an acceptable compromise.
I honestly think part of the reason the Doctor says that, and why the story starts out like that, is that people can easily be taken advantage of in similar situations. The MC is a well known person in the world, who is very well off, making that an even greater danger.
It's not entirely out of the question that it could be a concern - just because someone is family doesn't mean they're necessarily good people. I've actually informally disowned at least half my family, and I wouldn't want them anywhere near having access to my medical information or treatment decisions. I certainly wouldn't even remotely trust them to have access to me at a time when I was mentally impaired and potentially open to being manipulated or exploited.
That being said, the way it's presented in the story it is somewhat irresponsible on the part of the doctor, especially because he alludes to a lot of things but refuses to actually tell you anything straight out, in a deliberately and frustratingly obtuse way that is almost guaranteed to make things worse, not better. At a time when you are almost entirely helpless and confused, almost everything the hospital staff does seems calculated to confuse you even worse. Even the officially appointed therapist is absolutely terrible at her job.
(part of what was reinforcing my early suspicions that they WERE doing it deliberately, and that the story was going to devolve into you being manipulated and lied to at every turn)
But it's obvious from most of the rest of the game that the point of the story isn't to deal with how the medical establishment handles severe amnesiac cases as much as to fast forward past the minor details so the story can get you back home and interacting with your family. So most of the early stuff is just glossed over and half-assed to explain why the hospital is letting you go rather than keeping you in some sort of facility, and then it's all mostly ignored once you get home and the real interaction starts.