DrMirakle32 kindly hid the scene details under a spoiler tag in the OP.
tdalton is quite right that it's Mathis' death scene, in which Mathis reveals that his name is in fact not his real name.
It's because Bond asks Mathis about his name that suggests Bond has reason to think it's not real, ergo it suggests a common practise of using a fake name.
Therefore there's the subsequent suggestion that 'Bond' may well be such a name - one given to him by MI6, rather than his late parents.
So, could 'Bond' be an MI6 'hand-me-down' name?
One sincerely hopes not.
ETA:
In Skyfall is the family home referred to as the 'Bond' family home? I'm sure albert finney called him James, but did he use the word 'Bond'. If so, then it seems Eon has decided not to make Bond a 'codename'. I'd have to watch it again, but it seems odd to have the Bond ancestral home in the story, along with an old family employee who calls him James, if Eon are intending to make 'James Bond' a codename when it's time to replace Craig.
Well, actually, you even see Bond's parents' grave in SF... So, unless one assumes that MI6 also give codenames for dead people, we can safely say that the "codename theory" is once and for all pretty well burried (no pun intended, of course!).
Finney does say Bond as I recall.
Finney- "James? James Bond
Bond- "You're still alive?"