TRACON Controller reported they allowed an IFR aircraft to descend below MVA when the controller mistakenly thought they were VFR. Pilot reported they had terrain in sight and cancelled IFR.

Narrative

Aircraft X was inbound from west of ZZZ at 160 IFR; checked in at 160 direct ZZZ1 (his destination.) I told him to maintain VFR at or above 105; the pilot read it back. about 2 mins later the pilot questioned 'maintain VFR at or above 105?' I replied affirmative; pilot then stared a gradual descent. My assist controller said hey give that guy a terrain alert he is below the MVA. At which point I realized I thought he was VFR the whole time when in fact he was IFR. I Then instructed the pilot to climb to 150 immediately; MVA in their area was 144. They were at 138 when this occurred. The pilot read back climb to 150; you gave me maintain VFR about 105. To which I replied yes I did I apologize I thought you were VFR. The pilot then responded I have the terrain in sight and can maintain visual. I told him to maintain visual with the terrain but still need him to climb to 150 IFR. The pilot then said I can cancel IFR but maintain flight following. At which point I told him cancelation of IFR received maintain VFR. The pilot acknowledged. We do not normally get IFR aircraft going to ZZZ1 from the west; I was busy with saturation of other aircraft inbound to land at ZZZ and I failed to recognize this aircraft was IFR. Once it was noticed I did all I could to fix the scenario. Next time take the extra second to review the data block more thoroughly to ensure aircraft is in fact IFR or VFR to proceed then accordingly.