If information-theoretical and statistical mechanics entropies are NOT the same (or at least, deeply connected) then what stops us from having a little guy[0] sort all the particles in a gas to extract more energy from them?
Sounds like a non-sequitur to me; what are you implying about the Maxwell's demon thought experiment vs the comparison between Shannon and stat-mech entropy?
[0] https://en.wikipedia.org/wiki/Maxwell%27s_demon