If you give a nametag with a name to a fish that is currently swimming around, and you kill it afterwards, it won't have that name when it drops as an item. It will just have the normal item name, for example ''Raw Cod''.

This is not a feature request, because as of recent 1.20.3 snapshots, non-living entities such as boats display the name in both item and entity form. This is the same thing in my view.

As of more recent 1.20.3 snapshots, arrows now also keep their custom name after being shot and picked up (aka going from item form to entity form, then to item form again)