The '23' prefix appeared with the 'S' contracts, not just on BSAs and in fact not just on motorcycle contracts. It's shown on some Spare Parts Lists covers so it was clearly significant.
'34' is the Catalogue Reference Number and indicates that it was the 34th War Office contract awarded to BSA in the modern era (i.e. after the early 1920s as it doesn't included any Great war vehicles). RAF and RN contracts are also not included within this scheme. It seems to have been in intended to ensure that the correct Spare Parts Lists were consulted but this isn't so significant on most of the motorcycles as there was so little that changed...and the parts lists were rarely up-to-date with the newest revisions anyway.