Good question. Theoretically, yes, both would burn the same, given that the heart rate is the same. Heart rate increases due to an increased need to intake oxygen and excrete carbon dioxide. The need for oxygen intake and carbon dioxide out gassing, is driven by four parts of your cardiovascular system; heart, lungs, blood, and muscles. Without going into unneeded detail, depending on how each of those components are able to uptake, transport, and utilize oxygen will determine the demand placed on the heart to increase the rate of blood flow around the body. If one of those components becomes really efficient, then the demand for blood flow (oxygen) is decreased.
Lets speak about the muscles individually for a moment, as Joanne has alluded to. If you are a great cycler and poor swimmer (and hardly ever swim), then you will have a higher heart rate at the same workload for swimming than for cycling. This is due to ONE, increased movement efficiency, and/or TWO, increased oxidative capacity of the leg muscles, compared to the shoulder and arm muscles. I will note interestingly however, that cycling doesn’t show the same reduction in H.R. due to increased movement efficiency as does running for example. This is because cycling is unidirectional, performed in a single plane, and mechanically non-variant. So other than oxidative changes of the leg muscles, which primarily is the reason for a lowered H.R. response over time, there are minor movement pattern changes possible that could account for any significant H.R. drops during the same period. Swimming however, would show a significant decrease in the demand for oxygen over time, at the same workload (stroke rate), due to both oxidative changes and movement pattern changes. This occurs in swimming since individual movement patterns for that sport can vary greatly from what is mechanically optimal. Thus, improvements in movement efficiency can be greater and can more significantly impact oxygen demand, and thus H.R. over time. I hope I am saying it in a way that is not confusing.
But getting back to your question, I think there is a much larger issue that you might be overlooking. Who cares which burns more? Really, who cares? Meaning, the most important factor in helping a client decide which is the best cardiovascular exercise for them shouldn’t be based on which machine might burn more calories. In other words, if I tell a client to do machine X because it burns more calories, and they happen to hate machine X, then what I really did was tell them the worst machine to do. Thus, over time, they will do less cardio and burn less calories. However, if I asked them which machine they want to do, and they picked machine Y, then I would show them how to use that machine and possibly even ways to do it differently making it more difficult, etc. By focusing on the machine(s) that they like, I will get them to do more cardio, stick with their training program longer, and achieve more success with their fitness program and goals. Sometimes a clients personal psychology simply trumps any quantitative oxidative calculations that can be made. One caveat here, make sure the cardiovascular machine that you and your client decide on works well with their joints, and/or medical limitations, etc.