According to the GRO Index, there were 6 births registered in England and Wales as Patrick Graham 1876+/- 2 years - though only 1 registered in Newcastle upon Tyne District
Q1 1874 Gateshead mmn Cranney
Q1 1875 Pontefract mmn Hanbury
Q4 1875 Newcastle Upon Tyne mmn McQuire
Q1 1876 Liverpool mmn Power
Q1 1877 Whitehaven mmn Flin
Q1 1878 Leeds mmn McHale
I can't say for definite that he was enumerated twice but given the other occupants in both households, it does seem to be the most likely possibility.
Did your contact give reasons (references/documents) for saying that two entries mean they are definitely not the same person?
If you have doubts then take those births in the GRO, no matter where geographically they were born and using census returns/ deaths for these children prior to 1901, marriages for possible parents using Graham and the relevant mmn - etc etc, to see which if any you can discount.
As a starter
the child registered in Gateshead in Q1 1874
there is a likely death registration in Q4 1875 Gateshead for a child of this name who was aged 1 year.
and this wee one was buried at GatesheadEast on 5th Oct 1875, his Dad's name was Alexander
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSJ7-L95N-S?I looked for other births in Gateshead surname Graham, mmn Cranney and using those results and searching the 1881 census this looks highly likely to be the family at 37 Salt Meadows Tce, Gateshead
- bless them they had another go at producing a Patrick c1879 after the first one died.
Piece 5038 Folio 82 Page 22
So if your contact can't come up with something specific to back up what they are saying I'd suggest that you use similar methods to see if any of the others can be discounted. Its time consuming, a real pain sometimes, but firkling like this is my usual last ditch option to try to get to the bottom of a puzzle.
Boo