Shapeshift forms are statuses, not characters or something like that. The FANE tag (and any other origin tag) actually gets explicitly removed by script when you shapeshift. By default, it would just stay there.
You can easily re-add it in Osiris though, because at the same time, shapeshifting sets the SHAPESHIFT_FANE option.
Safest would probably be to also add the FANE tag again at that point. That may make some dialog options show up twice (in case there are separate ones for FANE and SHAPESHIFT_FANE), but it will avoid issues with script code that depends on SHAPESHIFT_FANE being set when Fane is shapeshifted.
So something like this should work:
IF
ObjectWasTagged(_Player,"SHAPESHIFT_FANE")
THEN
SetTag(_Player,"FANE");