In 1871 I can see two separate families.
1. John Wilson (63), Henrietta MA Wilson (his wife age 50) and 4 sons - John (12), Thomas (10), Richard(7), Arthur (26, a sailmaker). John was a mariner; everyone was born in Sunderland except Henrietta (London) and Arthur (Shrewsbury). All in Sunderland.
2. in Shrewsbury -- Arthur Wilson, a landagents clerk, lodging with the Felton famiy. This Arthur is also age 25 and born in Shrewsbury. I suspect he's with his parents in 1861 -- William and Eliza,
So neither of these fits the description of your Arthur with parents called Arthur and Mary!
Edited to add --- Just found a 16 year old Arthur with younger siblings and his widowed mother Mary in Sunderland in 1861. Did Mary remarry John Wilson thereby causing confusion?!