The incident occurred in just the second minute of the Premier League contest at the Etihad Stadium, when Wilfried Bony was fouled by Craig Dawson only for McAuley to get given his marching orders after play had initially been allowed to go on.
A statement from Professional Game Match Officials Ltd, the body responsible for refereeing appointments, read: "The referee has confirmed the offence was caused by a different player, which should be addressed now as a case of mistaken identity. The referee has apologised for his error."
City went on to win the game 3-0 with Bony, Fernando and David Silva on target, but West Brom boss Tony Pulis felt key decisions went against his side.
"If they're saying they're the laws of the game then the lad has to be sent off - fine," he told BT Sport. "But how he gets the lad wrong is just absolutely amazing.
"The second goal shouldn't be allowed, Saido [Berahino] gets kicked in the face. It's a decision that leads to the second goal and kills the game.
"It was going to be tough here 11 against 11. Saido's missed a great chance to make it 2-1. It was a game that was decided in the first couple of minutes."
Swarbrick's error is the latest in a string of high-profile refereeing mistakes in England's top flight.
Roger East also recently appeared to have sent off the wrong player when showing Wes Brown a red card for a tackle made by John O'Shea at Manchester United - though the official later claimed to have spotted an infringement by Brown.
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