For what it is worth I do not think that there is a kinship connection between the 2 Daniels.
Daniel born Sheffield - parents Joseph Gavin and Mary Ann Smith was baptised , as were his siblings, as a Roman Catholic. There is a possible birth for Joseph in Roscommon .( There is a West Yorkshire prison record from 1885 for a Joseph Gavin of Sheffield b Roscommon age 47 who was given 7 days for not sending a child to school).
Daniel born Salford -parents Alexander Gavin and Rosetta McKeowan were from co. Down, N Ireland and were protestants. They married in Glasgow and their eldest son Alexander was born there. ( Criminal records show that this Daniel had aliases as well - he was known as Daniel William and William Daniel!).
I wonder, Vendee, if it might be worth contacting Lancashire Archives as it would appear that the records for Lancaster Asylum are housed there.
William