Caffeine == Caffeine
But if a Coke is ~3mg caffeine 100ml and coffee is ~40mg per 100ml, drinking 13.3 "units" of cokes will not be equivalent to 1 "unit" of coffee.
1. Timing of the intake will affect the efficacy (x mg of caffeine over 2 hours vs. 20 minutes is very different to the body)
2. Controlling for time-dependent intake (the spherical human consumed both quantities instantly), concentration of the substance affects the rate / efficiency of absorption by the body (I cannot predict for which way it will go, this stuff is complex) so the net absorbed caffeine will be different between the two substances
3. Delivery mechanisms are different: Coke is very sugary, black coffee is not and is a colloidal suspension. The local chemical environment for the caffeine within these two substances will be different. Sugar is known to affect osmosis, meanwhile who knows wtf the gunk in coffee is doing at the chemical level.
So to sum up "caffeine = caffeine = caffeine" is probably a useful rule of thumb and will get pretty far, but at the physiological/biochemical level, that is as useful as armchair pseudoscience is.
To use a possibly relatable issue: most people find consuming shots of vodka more efficacious than consuming the equivalent amount of beer by alcohol. 3 shots will probably get you more wasted than necking 3 beers