It's not an act at all. He's clearly never tried to come off as even a little relatable to the average British voter, which I guess makes his uber-posh identity at least a little bit of a choice, but I don't think he can imagine being anything other than the son of a Lord and the father of a child named Sextus or whatever.
I'm also shocked the man is a Catholic, though. I know some very old lineages in England managed to keep Catholicism but still
The thing about Rees-Mogg is that he’s in no way an aristocrat - his father wasn’t even raised to the peerage until Mogg the Younger was at university, and his grandmother was an Irish-American Catholic (which I assume explains his Catholicism). The name Rees-Mogg, in spite of being double barrelled, isn’t indicative of any aristocratic lineage. He’s essentially self-consciously old-fashioned, although he seems to have been this way for so long it would be wrong to call it a persona at this point.
I’ve never really seen anything wrong with it myself, although it seems to trigger some people on the left (and some people from genuinely blue blooded backgrounds) considerably. He’s not exactly the first British politician to have adopted a persona largely at odds with his ancestry and social background (see Jenkins, Woy and Benn, Anthony Wedgwood).
His father's family did belong to the gentry and got the double barreled surname when they took over a manor.
https://en.wikipedia.org/wiki/Cholwell,_Cameley
