Quite possible that Archer was what was heard by the registrar, not too dis-similar given an accent, and possibly not noticed if Lily didn’t sign her name (not looked at the images).
Only last week I met a chap with the surname Archer who had traced his paternal line (very well documented) and hence surname back five generations to an Orchard. Orchard and Archer are variants of the same name. As Deirdre says it just takes a mishearing for a name to be written differently and then stay that way.
Can I add something else into this one - any suggestion that the Orchards were from the Travelling community as Orchard is a recognised surname from the community?
Lily's surname was Hill, but it looks as though the Registrar filled out the entire form including Lily's name. It is possible they weren't very literate. However I could not find a Sophie or Sophia Archer marriage (by checking UK free bmd) that seemed likely and had any daughters (by checking UK GRO for husband's surname and mmn Archer/Orchard). I am led to believe the mistake was intentional. There is no James Orch either. There is a James Arch but the Sophie/Sophia Orchard/Archer marriages do not show an Orch spouse.
There was indeed a Travelling Community of Orchards. Sophia Orchard b. 1857 Torrington Union, was part of that community. Sophia's (b.1857) father was James Orchard. In at least one census they were living in tents.
Sophie Orchard's (b 1858) father was Richard Orchard, a rope maker in the Royal Navy. Richard is recorded as Sophie's father in Sophie's marriage to Thomas Bowdidge in 1887, after the birth of Violet which was in 1884. In the 1881 Census Sophie's father, Richard was away at sea.
Sophie's mother died in 1871. Sophie's father Richard re-married in 1879. Sophie's sister, Lily, married W J Hill in 1880. In the 1881 census Sophie, her father and her siblings, although not living in the same house, were all living in Plymouth. In 1884 when Violet was born Sophie and Lily were living in Plumstead, which is I believe, like Plymouth, near Royal Navy Dockyards.