"Flat" increments "number" if and only if it has a separate post box. If you have several flats behind the same box in a common front door - a common set-up in the UK when a larger house has been converted into several flats - then as far as the postcode system is concerned, those flats don't exist; since people still write "Flat 3, 11 Wisteria Drive ..." on letters this creates various issues with denormalised addresses.

How much influence does a building owner/developer have over this assignment? Can they explicitly request multiple post codes for a building?

Not always, I've had that exact situation and the flat number was in all systems. All houses in my postcode had several entries for flats and as far as I can tell, they all only had one box.