Tower Controller reported a departing aircraft deviated from the departure procedure and flew below the minimum vectoring altitude.

Narrative

Aircraft X was issued the ZZZ1 DP (Departure Procedure) by Clearance Delivery. I taxied the aircraft to Runway XX and then cleared them for take-off. When airborne I told them to contact departure; but a moment later; I saw them in a left downwind turn and were well below the Minimum Vectoring Altitude (MVA). I reached out and they were still on frequency. I verified they were on the ZZZ1 DP; and the pilot said something to the effect of 'Oops'. I issued the heading for the DP and restated the interim altitude. After observing them on the correct course I again told them to contact Departure. These aircraft typically file IFR but don't pick it up and instead depart VFR most of the time when the weather is better. Today we had lower clouds so they picked up the IFR [clearance]. They are so used to picking up VFR departure instructions that my guess is [that] expectation bias played a role in the pilot just starting a southbound turn.I wouldn't have given this much thought except just yesterday I had a similar interaction with another flight on an IFR DP. I think it may have actually been the same pilot but I can't be sure. They were cleared for take-off on [Runway] XXL on the ZZZ1 DP; which would have kept them runway heading and; off the departure end of the runway; they asked me if I wanted them to start their turn early or fly to ZZZ1. I verified they were on the ZZZ1 DP and told them to just fly it as they were previously cleared. The way they asked the question made me concerned that they thought they were VFR.I'm not sure if I have much of a recommendation for this scenario except that maybe some outreach to [their] Company might be helpful. Having two incidents back-to-back where there seemed to be confusion about VFR versus IFR status makes me concerned that there could be more. I just wanted to get these two on record in case there are future problems. A report was entered for the one today.