The description in the pico-sdk documentation describes the irq_set_exclusive_handler function as:
Set an exclusive interrupt handler for an interrupt on the executing core.
It was clarified to me on the forums that the current SDK only uses a single interrupt vector table (VTABLE) for both cores, so as a result this function effectively sets the exclusive interrupt handler for both cores. Until/unless the SDK is updated to support separate VTABLEs per core (#2158), IMO this function description is a bit misleading as written.