FIR is an ICAO concept relating to civil aviation. Military operations do not have to follow ICAO rules and FIR procedures, never mind request permission to enter a FIR zone in international airspace.
Even in regards to civilian aviation FIR airspace is not subject to the whims of the nation administering it (unless the airspace is also in their territorial airspace). Airlines can be banned for territorial airspaces not FIRs.
So no Incheon FIR is not internationally recognized as South Korea airspace.
Nor was the KADIZ set up to support Inceon FIR it was setup during the Korean war by USAF.
Having said that pretending to have an ADIZ is not illegal in itself as long as you do not take illegal actions in support of it. You can track planes via radar,ask them to identify themselves (they can ignore you) and even fly up to them in international airspace.
Though shooting even warning shots as they claimed they did in 2019 would be an illegal act of agression.
You do not need to proclaim any zone to prepare for an enemy attack. The only need use of claiming an ADIZ is to propagandize that you are being 'violated' by perfectly legal behavior.