The 1851 Census returns show Robert Fletcher as born in Kelso, Roxburghshire, and his wife Euphemia as born in Aberdeen.
Robert Fletcher, who was living in The Pleasance (a part of Edinburgh) at the time and Euphemia Christie, daughter of James Christie in the West Port (a part of Edinburgh) gave their names to the Session Clerk in order for the banns to be proclaimed in church. The Session Clerk recorded this under the date of 4 September 1803 (OPR 685/2 Register of Marriages, Parish of St Cuthbert's, Edinburgh). As was common, the date of the marriage itself was not recorded. Euphan was just an old pet form of the name Euphemia.
The 1851 Census returns show a James Fletcher, 33, with no relationship descriptor, in Robert and Euphemia's household in Leith. James, however, appeared to be their son. The OPR Register of Births for Leith North show that a James Richard Fletcher was born 15 August 1817 and baptised 21 September 1817 to Robert Fletcher, shoemaker in Citadel Street and Euphemia Christie
There was a Peter in 1811, also in Leith.
If Robert's mother's name was given as Margaret Christie (Christie?? - same maiden surname as Robert's wife? - certainly not impossible, but it should set alarm bells ringing, just in case the informant did not know with 100% certainty what Robert's mother's maiden surname was but did know that there was a Christie name associated with Robert and the informant thought that that must have been Robert's mother's surname. I'm not seeing a P Fletcher (just use "Name starts with" option, to cover Peter and Patrick) and a Margaret anything marriage anywhere in the expected part of the world around the probable time.