Yakin questioned the referee's decision, calling it a "harmless foul" and saying the rule "destroyed our game today."
He added that he was unaware of the specific protocol used by VAR officials before the incident.
Swiss captain Granit Xhaka acknowledged the red card changed the game.
"This decision was difficult to accept now after the game because the dressing room was very quiet, disappointed," he said.
Argentina coach Lionel Scaloni admitted the numerical advantage played a role.
"Because of a red card, one of their players was sent off, luck was on our side," he said.
This was the second instance of the mistaken identity protocol being used to overturn a yellow card during the World Cup.
The regulation allows VAR to intervene when an incorrect player is penalized.
The controversy follows similar complaints from the Egyptian Football Association after their Round of 16 loss to Argentina.
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FIFA's chief of refereeing, Pierluigi Collina, defended the officials, saying, "Nobody can question the integrity of the FIFA World Cup match officials."
