Green’s Dictionary of Slang

to light a cigarette (for someone).

1952		J. Jones From Here to Eternity (1998) 162: Gimme a butt before I go [...] Now match me and I’m gone.
1957		E. Lehman Sweet Smell of Success [film script] Match me, Sidney.
1999		Observer Screen 26 Sept. 6: Burt Lancaster holds up his unlit cigarette and contemptuously orders Tony Curtis to ‘Match me, Sydney’.