No it does not need to be on a N registered aircraft.
(1) Accomplished a flight review given in an aircraft
for which that pilot is rated by an authorized
instructor;
that`s it...
In South Korea, Japan, China where there are a ton of FAA CFIs they do BFRs on local registered ACs...
When I was at SUSIAIR, the FAA rated pilots would do their BFRs on the Caravan with an FAA CFI....