The census evidence seems to point to Ann Bailey being illegitimate. When illegitimate people married, if they could get away with it, for proprieties sake they often fudged their father's name and made sure he certainly had the same surname as their unknown. The first name they gave him might be correct or if they weren't sure they made up a name, usually one from within their family circle. Likewise the occupation they gave their father.
Information on marriage certificates should always be checked against other sources because basically it can be very unreliable. There were no checks in the system and unless you were known to the vicar or the registrar you could say whatever seemed reasonable with no-one questioning whether it was correct.
The censuses seem to indicate (unless you have found another Ann Bailey born Fordingbridge of the same age) that Ann's mother was called Mary and was unmarried. Ann's birth certificate (or baptism) would prove this one way or the other.
Other than what Ann chose to say on her marriage about her father have you any other evidence that Ann Bailey's father was actually called George Bailey?
This in the baptism of what from the census information appears to be Ann Bailey's older sister.
Matilda Ann Bailey
Christening: 31 OCT 1841 Fordingbridge, Hampshire
Mother: Mary Bailey
and a possible for Ann's birth. The certificate itself would give further details.
Births Jun 1856
Bailey Ann Fordingbridge 2b 512
I think this is the George and Martha Bailey in Hampshire (but not Fordingbridge) you expressed an interest in
1861 census RG9 668 folio 97
Ringwood Hampshire
George Bailey 38 Head Married Farmer of 12 acres
Martha Bailey 32 Wife Married
Eliza Ann Bailey 4 Daughter
James Henry Bailey 2 Son
Charles Bailey 2 Months Son
all born Ringwood
By 1871 they had moved to Hordle where George was a farmer of 22 acres and had a further children three children all born Hordle.
Alice 6
George 5
William 1
On the 1881, 1891 and 1901 censuses in Fordingbridge Annie Sainsbury nee Bailey gave her birthplace consistently as Fordingbridge.
Regards
Valda