Thank you for the nice explanation, despite my somewhat rude and uninformed posts! I had some trouble accepting it, but the heat/work example made it "click" for me, once my brain got past its initial stubborn reaction of "but of course heat and work are both energy, whereas mass is clearly not!"

So any time two physical quantities are related by a constant factor -- even if said constant factor has what we currently think of as "units" (and even particularly weird units like squared speed) -- they are fundamentally equivalent. I'm still mulling this over, but it's a pretty fascinating mental revelation for me.