The Reds were comfortably beaten by Los Blancos in the first leg of the Champions League quarter-finals.
Vinicius Junior had opened the scoring, before Marco Asensio doubled Real Madrid's lead in the 36th minute as Mane appeared to be fouled by Lucas Vazquez in the build-up to the second goal.
The Liverpool attacker was booked for dissent three minutes later and Klopp was not happy with his player's treatment.
He said: "The situation with Sadio, what the ref did tonight I have to say I don't understand.
"For me that was something personal because he dealt with the situation with Sadio, which was a clear foul, like he was a diver or whatever.
"From that moment on whenever Sadio went down he didn't get anything. That is not right. That is what I told him after the game, that I thought he was unfair with Sadio.
"That doesn't change anything at all. He [Brych] didn't lose the game. We were not good enough to get a better result, but in these moments you need just an 'okay' ref. That would have been enough."
Klopp admitted that Liverpool deserved to lose on the night as Mohamed Salah's away goal gave them hope of rescuing the tie at Anfield.
He added: "First and foremost I don't think we deserved to win tonight, we didn't play good enough and that is my first concern.
"Especially in the first-half we didn't play good enough football. We conceded the first goal, served the second on a plate, but that was not the real problem of the game.
"Of course conceding goals is a problem but it is not the problem. The second-half we did better, we scored our goal which is the positive of the night and that's how it is.
"Did we deserve more than the 3-1? I'm not sure."
Jurgen Klopp believes referee Felix Brych had a personal issue with Sadio Mane as Liverpool fell to a 3-1 defeat against Real Madrid on Tuesday night.