Does a receptacle have to supply the max current that it is rated for?
The plug specs are the maximum ratings given for how the plug is meant to be used in any given application. So a plug speced as 120V/30A is not designed to be used in applications over 120V and not to be loaded over 30A.

Now that said....those numbers do not directly have anything to do with the capabilities of the supply circuit or wiring. If the circuit was wired and deployed with the proper National Electrical Code guidelines then yes a 30A outlet would be wired with proper sized conductors capable of supplying 30A and the distribution panel being equipped with the proper sized circuit breaker rated at 30A.

However there is no guarantee that the distribution panel can always deliver the full 30A to this outlet. It is possible that the panel can support 100A total via its main breaker but there could be an active kitchen range drawing 50A on one circuit, an AC unit drawing 30A on another circuit leaving only 20A available. If you put the full 30A load on the outlet under consideration the main breaker would trip.

Regarding the voltages....outlets designed for 120V will generally be rated for supporting circuits at a nominal range of 110 to 120VAC. Similarly for outlets designed for 220 to 240VAC. The contact arrangements will be different for the two voltage ranges to prevent folks from attaching devices to the wrong voltage.