Referee Paul Tierney only showed Kane a yellow card for the challenge but later showed Robertson a straight red card for a tackle on Emerson Royal.
Liverpool manager Jurgen Klopp was angry with some of the decisions that went against his side, who also saw a penalty appeal for a challenge on Diogo Jota waved away.
It has been pointed out that the decision to allow Kane remain on the field was because Robertson took his standing leg away from the tackle but Clattenburg, who spent 13 years as a Premier League referee and took charge of the European Championship and Champions League final in 2016, does not agree with the decision.
"To hear that Robertson has to have his leg planted alarms me, because if he does that he's not walking this Christmas," Clattenburg told the Monday Night Club on BBC Radio 5 Live.
"If you don't believe this is a clear and obvious error about Kane, you're not doing your job correctly.
"I think we, as referees, are sometimes guilty of knowing the laws of the game but we don't understand the game.
"VAR cannot get this wrong. Referees can, because they have a split second. VAR have all the angles, he's got all the slow-motion, he can see the point of contact.
"If they are saying that his (Robertson's) leg has to be planted, which is a new one to me, if the leg was high, the studs were showing, it's reckless, he's lunged. For me, he's endangered the safety of the opponent.
"Robbo is lucky today that he's still walking. We should understand footballers more because he's not going to leave his leg there, why should he? He's not going to want his leg broken and his career put in doubt."
