Flight Instructor with student reported confusing ATC instructions resulted in an NMAC requiring both aircraft to take evasive action to avoid a collision.

Narrative

My student and I were climbing out after takeoff from GEU and attempting to make a northbound departure. We were cleared for takeoff and advised to fly 'straight out.' As we were reaching approximately 2500' MSL; the tower controller advised something along the lines of 'maintain your climb at your current altitude.' My student and I were unsure if the controller meant for us to maintain our current rate of climb or maintain our altitude. We continued our climb during the next several seconds as we were trying to determine what the controller was asking of us.There were other transmissions occurring over the next several seconds preventing us from clarifying; And the next transmission from the controller to us was a traffic advisory of another aircraft at our current altitude (now approximately 3000' MSL) going the opposite direction at the same altitude. We saw the other aircraft and both our aircraft and the other aircraft took evasive maneuvers to avoid a collision. We were then cleared to proceed with our northbound turn.