As you have found, unnamed children are indexed on the GRO on-line index with both a blank surname and forename. Why they don't use the same indexing rule as other entries and show the surname as that of the the father/mother ....I've no idea.
That's something I didn't know. Thank you Antony.
Although I have two children whose records contradict that, one un-named male and one un-named female, both died within 24 hours of birth. Both have a single dash in the Name column in their birth entries in the register, and a " - Merriweather" or " - Merryweather" in the Name and Surname column in their death entries.
They are in the online index as:
Births: "MERRIWEATHER, - " (1883/Q1/West Ham/4a/75) and "MERRYWEATHER, - " (1884/Q3/West Ham/4a/96)
and deaths: "MERRIWEATHER, - " (1883/Q1/West Ham/4a/32) and "MERRYWEATHER, - " (1884/Q3/West Ham/4a/54)
But your information has prompted me to search again for another rumoured child that I have so far been unable to find any trace of.