It makes sense that Aqua would hide his past identity after Ai died and he started down his dark path of revenge, but prior to that it makes no sense that neither of them ever discussed anything about their past lives. They literally spent years only able to have real conversations with each other but, apparently, they somehow managed avoided ever talking about hospitals, cancer, doctors, etc.
I initially saw that as this story's biggest flaw but accepted it as a sort of artistic flair that would never be acknowledged like a reverse Chekhov's gun, so I have some really mixed feelings about this chapter. It either should have happened way way earlier or not at all/not until the end (wait... what if the story is ending. That'd be funny)
It actually makes sense to me that they never talked about it as babies. They were four when Ai died, and neither of them really wanted to think about their past lives and regrets. Ruby/Sarina had a pretty miserable life in the hospital and a lot of unresolved trauma (as shown by her reaction to seeing her original mom), so it's natural she wouldn't want to talk about all that immediately. She embraced the fact she got a second chance with her idol, and seemed to be trying to ignore her past life as a coping mechanism.
While Aqua/Gorou had a happier life, he also strikes me as more reserved in general. And then he got murdered while tending to Ai as her doctor. So bringing up either of those facts out of the blue would probably feel weird to him. That, and he probably didn't want to think too much on how he was mentally older than his new mom.
Also, it's actually really easy to avoid talking about hospitals, cancer and doctors when they don't play a significant part in your current life. I doubt a routine checkup would be enough to get them to fall into a deep state of reflection and start unloading their traumatic pasts.
If it wasn't for the murder, I think they probably would've talked about their pasts around the time they started school. It's a neutral topic to reminisce over, and it's the earliest natural opportunity for Aqua to mention he was a doctor through a comment about going to med school ("this will probably be pretty boring after med school").