I don't think evidence of identity was needed then, the Registrar simply took the details given to him by the informant as far as I know. hence such things as age at death depended on what the informant thought!
It may be a good idea to obtain the marriage certificate of Benjamin Wright senior to a Martha Fisher which occurred in 1871 --- Dudley Registration District. This should give her father's name, which will be evidence of her maiden name. (I suspect Fisher to be her first married name).
Also on the marriage cert you have for Benjamin junior -- does it give his father's occupation as Boatman or Waterman or the like please? If it does, that will be more evidence that the Benjamin Wright on the 1881 census in Runcorn, with wife Martha and several children - is the correct father (or who he thinks is his father.)
Like you I am struggling with this case -- if the family were travelling on the waterways, it may just be that a formal birth registration was not made -- but Martha's actual maiden name from her marriage cert to Benjamin senior in 1871 would be very useful.
(The person with Martha Fisher, widow in 1871 -- shown as 'Mate' was a William Beresford -- and there is a William Beresford, brother of both the Caroline and the Martha Beresford on the 1851 census -- but this MAY just be a coincidence.)