What is the appropriate unit for measuring power consumption by a section of code?
Energy consumption measures energy. Full stop – if the paper measures anything else than energy, it's mislabeling it.

Energy has the SI-derived unit J (joule), which is 1 J = 1 W·s; the rest is trivial conversion of units. I don't know what to explain about that – having a clear understanding of what power is, what energy is, what current is, is probably necessary anyway, if you want to understand that paper.