Well said, one thing to add is it's not uncommon to have the CPU disable further interrupts as part of the automatic handling. Or it may be configurable, it's often not something the interrupt service routine needs to explicitly do; although, sometimes it is, there's so many options.

If you're on a system where interrupts are disabled automatically, and you actually do want re-entrancy sometimes, you can usually make that happen too, but you might wait until you get to a safer place (maybe switched to a kernel stack or ??? again, so many options)