@throwaway100000 summed it up beautifully:
In my experience born and bred Londoners understand why, gentrifiers/outsiders don’t
Dual identity is normal when people immigrate from another country, it's not a difficult concept. Large numbers will say they're from Bromley in Kent, but if asked where that is will say East London, there's no harm either way. The difficulty isn't with accepting a link with Greater London, it's with being asked to give up a cultural element of one's life and to become a faceless "outer" place to the main event of London instead of retaining a sense of the local history of the actual place.
In many of these outer boroughs there is a sense of place; the names of the place and other counties appear on shops, sports clubs and old addresses, there are local museums and history groups, so the connection is gently built from childhood. If you grow up going to Brownies/ Girl Guides in Middlesex while supporting their cricket team (Chiswick, Wembley, Harrow, etc), or if your town pre-dates the 1086 Domesday book as a significant market town (Croydon, Sutton, Kingston upon Thames), if you have Surrey County Council's County Hall in the middle of your borough (Kingston upon Thames; also problematic because it's a Royal borough rather than a London borough), if your market town was chartered in 1158 in Kent (Bromley)... then that is what describes your home. There's no need to suggest people must sever any connection with a place, actually it's rather unkind to try.
Inner London areas either have their own strong local history (Brixton, Camden) or are part of the fabric of London as a whole (Victoria, Westminster, the City); it isn't surprising that they should feel Croydon isn't "real London", because they share the present, they don't share the history.