The problem likely lies in old ties from prior linkages somewhere in the software / communications database ether. But unless you factory reset every Apple device with any prior linkage, delete every contact with linkage, every old iMessage thread, every tie of prior AppleID possible linkage, restart on separate AppleIDs, etc., etc. it will be hard to know. A solution that has worked for our family (where we share AppleIDs across a few people) has been to (1) update to latest iOS release 12.0.1 (2) ensure one phone (and only one phone but you need one) is set up to "start a new conversation" from the AppleID email address in messaging settings (in our case it's my phone), and all others linked to same ID send/receive from the carrier phone # from their SIM card for the phone and (3) delete all old thread where the linked issues were taking place. This has worked for others in this discussion thread as well. Ours has been fine now for over a week.
In the for what it's worth category, I will repeat again one person's opinion (my own) since the only posts in recent days have been from the other side - Apple does indeed not recommend using the same AppleID for more than one human, that is true. It can lead to issues like these. It clearly was not designed at the outset for more than one human per AppleID. For a lot of reasons, but one simple financial one is people cheat and have 50+ friends and family "sharing" music and other content rather than paying for it.
That said, they know people do it, and several people I have spoken to in support at various levels understand why in the case of parents, etc. folks do it. As a parent of a child who has suffered with depression and anxiety, I would do it again and feel if this is what parents want to do, they should go ahead and do it - regardless of what others in this thread think. Is it subject to risk of issues such as this? Yes. Does Apple know it happens and do they try to "support" where they can - yes, they do. They do not recommend it, but they acknowledge it and will try to help you if you call - keep asking for a higher level of support until you reach someone who is senior enough to be both knowledgeable and a solution driven support person.