I think there must be lots of different reasons for using "otherwise" in this context, but possibly the reason Ann Littler was otherwise known as Watson is because her mother married, either for the first time or when she'd been widowed, after Ann was born. There was no formal adoption process in the UK until the 1920s, so changing a child's surname was voluntary before then.
John Catchpole otherwise John Catchpole Stanford could have been born illegitimately to a mother with the surname Stanford, who later married the father who had the surname Catchpole. If a couple had a child out of wedlock, but they were pretty sure they would later get married, the child was very often given the surname of the father as a middle name, so that the surname of the mother could just be dropped once the father & mother were married.
As I say, these are just possible explanations - there could be many more, but I hope it helps.