Based on experience with the decimal module, I think this would open a can of worms. To match what decimal does, we would need a Context() object with methods for dump, dumps, load, loads. There would need to be a thread-local or contextvar instance accessed by getcontext and setcontext, and perhaps a decorator as well. We would need a few pre-made instances for common cases.
Also, the decimal module was context aware from the outset. For JSON, we have large body of pre-existing client code that was created and tested without the concept of a context. Should existing code use the new context and possibly break assumed invariants? If the existing code had explicit parameters (such as indent=4), would the context override the parameter, take a backseat to the parameter, or raise an exception?