A TRACON Controller providing OJT reported the wrong aircraft checked on their frequency due to similar sounding call signs and was issued instructions meant for a different aircraft. The adjacent sector controller issued descent instructions to the other similar sounding call sign aircraft which placed it below the MVA.

Narrative

I was providing OJT on a sector when we took the hand off on Aircraft X; descending via the ZZZZZ arrival into ZZZ1. We issued him a heading of about 290 for the visual approach. Unknown to us; an Aircraft Y was south of ZZZ1 on the ZZZZZ1 departure. That aircraft was not on our scope. Aircraft Y had taken the frequency switch from another sector to us; instead of Aircraft X. To make matters worse; Aircraft X did not take the switch and was still following directions intended for Aircraft Y on the other frequency (XXX.X). Both controllers were working the wrong aircraft; and we didn't even know the other aircraft existed. Call signs were read as just the last three letters because no other aircraft was on frequency. Aircraft Y was supposed to be southbound and we inadvertently turned him westbound; thinking he was the other aircraft. Seeing Aircraft Y turning westbound; the other controller turned Aircraft X southbound; which was the ZZZZZ arrival who was supposed to be westbound. Aircraft X entered the 079 MVA at about 070 before the other controller figured out what had happened. We continually tried to turn Aircraft X back to the west; when actually we had Aircraft Y who was indeed west. Finally the aircraft were sorted out and got to their correct frequencies.Similar sounding callsigns phraseology might have helped the other controller keep the planes on the right frequency. If we had used the full callsign more than just the first time at the sector; it might've cleared things up earlier as well; though I'm not sure we had enough information to do that since we didn't know there was another aircraft anywhere with a similar callsign.