I think I can answer this to a degree.
Not sure but it may have been an AMATS proposal element in 1991 that when Flight Service ceased and functions transferred to ATC, directed traffic info to IFR RPT & military low jets in the then "OCTA" would cease (this predates going to the ICAO airspace classifications of ABCDEFG).
Naturally the Industry kicked up a fuss, and DTI continued, then when the ICAO classifications came in later it was decided to classify this "OCTA" to Class G, even though the provision of DTI (I think) is an element of Class F airspace.
As a result someone here from time to time has called our Class G "ForG".
I don't have ICAO SARPS handy but I think there is some other element of Class F airspace that means our Class G isn't fully compliant with the ICAO Class F hence it isn't called as such.
Edit: Definition of Class F & G airspace:
Class F.
IFR and VFR flights are permitted, all participating IFR flights receive an air traffic advisory service and all flights receive flight information service if requested.
Note.— Where air traffic advisory service is implemented,this is considered normally as a temporary measure only until such time as it can be replaced by air traffic control.
Air traffic advisory service.
A service provided within advisory airspace to ensure separation, in so far as practical, between aircraft which are operating on IFR flight plans.
Class G.
IFR and VFR flights are permitted and receive flight information service if requested.