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Can you also expand on why? Especially in this case I do feel like it’s an (imperfect) proxy for power they wield.

The same reason you don't compare speed with distance. Distance is meters, speed is distance per the unit of time, meters / seconds, m/s.

Likewise, market cap is measured in monetary units, dollars, euros, etc., while GDP is measured in money per the unit of time: $XXXX / year.

Because one is annual, and the other is just total. The equivalent of GDP for corporations would be annual revenue.