I'm replying before reading the manga so I might be taking this out of context but, if a person has already developed DID then it is possible for traumatic events that occur later in life to create new fractures or 'alters' to deal with the stress. DID doesn't usually go away as you get older (though alters have been known to disappear or reintegrate within a system), and it can change in many ways throughout a person's life.
Ok, so DID does result from severe childhood trauma, but symptoms usually present in adulthood. While children can be diagnosed with DID, it is difficult to do with certainty. So, it is usually diagnosed in adulthood. With the way his father treated him, he more than likely experienced a lot trauma in childhood that accumulated until the alter presented when his father tried to kill him. Basically, that last trauma was the trigger for the alternate personality to present itself.
Symptoms can present in early childhood, the average age to be diagnosed is 16. Less than 20 percent are diagnosed/present with symptoms after the age of 20. So is isn't usually diagnosed in adulthood at all. But its totally possible and for the sake of this manga it makes sense for the reasons you mentioned.
Correct me if I'm wrong but aren't they like adults now? I'm not that knowledgeable about DID but I know this only occurs during childhood. When the child experiences repeated traumas at the sensitive developmental stage (i think around 6-10???) Since that's the time children starts to create "personalities" and such.
So I was really confused when Yuki said "and Yuuta was born." I dunno, I'm not a professional but that part literally confused me... if you have any knowledge about this may you please explain it to me?