Or are you dividing the cost of that banking system by the number of transactions it processes? I.e. comparing 1 transaction to 1 transaction?
On a per-transaction basis traditional banking uses a lot less energy than Bitcoin. Including the cost to run the bank's servers, the air conditioning in their buildings, the Christmas lights their employees put up, and everything.
How can I make such a bold claim? How could I possibly have calculated the energy usage of all those things?
Simple. Bitcoin does about 8.5 TPS which works out to around 268 million transactions per year. Bitcoin uses 0.55% of the world's energy production per cbeci.org.
The ACH network (US direct deposit system) processes around 23 billion transactions per year. The European SEPA network does about 22 billion. The UK's BACS accounts for another 8 billion or so.
So just counting those 3, that's 53bn transactions per year, which is 197x what Bitcoin does. If their energy cost per transaction was as bad as Bitcoin's, then that would take 0.55 * 197 = 108.35% of the world's energy production. There literally isn't enough energy in the world for the banks to be as inefficient as Bitcoin. And we're only handling direct bank debit/credits in the US and Europe so far, we haven't even touched on credit card stuff, international wires, anything in Asia, etc.
That these things have any value attached to them is only due to people agreeing to it and believing it. Which makes them no different from seashells. While seashells would be a lot less practical in todays world they have been used in the past.