I think the most likely explanation for the talk of DID is that the very lay terms 'split personality' or 'multiple personality' are being used by WC's wife and her family in the course of trying to come to terms with what he's done, as compared with the family man they thought they knew. They are engaged in a kind of grief process and are having to travel at warp speed from denial to acceptance, and also having to find ways to talk to his children about it all. The idea that he was two (or more) very separate people because of a psychiatric disorder that isn't his 'fault' must be a very attractive narrative for them right now, and who can blame them.
I think the lay language is your clue that this isn't coming from a psychiatrist, and the guilty plea makes it clear that no one is expecting anything to come out that mitigates his actions from a psychiatric perspective. If it's true that he's working with psychiatrists, then I think it's probably, as
@Sherlockhames says, that they are trying to get him to accept what he did enough that he will share detailed information with investigators, probably in conjunction with trying to mitigate suicide risk.
JMO
[edited for clarity]