David, I am so sorry for your loss. <3
I think it may depend on where you are. New Zealand registers stillbirths, but there is no corresponding death register entry. Some places don't register the birth but do register the death. As with NZ, some places register the birth, but don't register the death. My great-great-grandparents had a stillborn child. Their only daughter. She had neither birth registration, nor death registration (that we (or the Queensland Registrar General's office) have ever found). All we have is a newspaper notice that the Judge's wife had a daughter, stillborn. That was Queensland, Australia, back in the day. Where there was a registration record, generally a certificate will be issued.
It is a tough area for some of us, so we do try to treat it with sensitivity. We give the babies a profile, because they were part of the family. They help tell the tale of the mother, as well. If it is not known if they were male, or female, then no selection is chosen. (For the standard WT profile image, that means it is half-and-half.)
What is harder for some mothers is the babes that are classified as "spontaneous abortions" aka miscarriages. The baby never got to be viable enough to be a possible birth. (This has been a tragedy more than once for my direct family.) There is not a one-off fits all answer.