A Center controller reported an aircraft departed a non towered airport on an IFR clearance erroneously issued by an adjacent facility and flew below the Minimum Enroute Altitude.

Narrative

Aircraft X called me in the air off of ZZZ airport. I asked them if they were requesting to open an IFR flight plan or if they wanted VFR flight following. Aircraft X responded that they had already been cleared IFR off ZZZ airport. I was confused by that response; becuase I had been on position there for awhile; and I had not cleared Aircraft X to take off from ZZZ; and I did not have a d-side; so no one else could have done it. I searched Aircraft X in the EDST (En Route Decision Support Tool); and there were no flight plans with that call sign. I asked Aircraft X what code they were assigned. I searched the code they said; and that code belonged to a proposal off of ZZZ1; and it was not Aircraft X.I did see Aircraft X ADSB readout; and they were at 028; below the MEA. Aircraft X advised they could maintain VFR if that helped. I told Aircraft X to maintain VFR at or below 100; and I and requested information like their aircraft type and destination. After I had enough information; I radar identified Aircraft X and cleared them to their destination IFR.Today; I was informed that on that day; Aircraft X had filed a flight plan departing from an airport 150 miles east of ZZZ; and they called ZZZ1 Center for a clearance off of that airport. The ZZZ1 Center controller gave them a clearance off of the other airport; but Aircraft X took off from ZZZ. This was an extremely unusual situation. I don't know how to prevent it from happening again; but it was so unusual that I don't know if it ever will happen again.