Brazil international Vinicius reported an instance of alleged racial abuse to referee Francois Letexier after scoring in Tuesday's Champions League play-off tie against Benfica, with the Spanish club later pointing to winger Gianluca Prestianni as the subject of the complaint.
Prestianni denied making racist comments in a post on Instagram, while his club spoke of a "defamation campaign" against him.
Letexier activated anti-racism protocols, suspending play for 10 minutes, with UEFA initiating an investigation into the incident.
Mourinho was criticised for his post-match comments, saying that "something happens, always" in matches where Vinicius plays.
Kompany addressed the issue at a press conference ahead of Bayern's Bundesliga match against Eintracht Frankfurt, and issued an impassioned defence of Vinicius, stressing his reaction "cannot be faked".
"For me, even worse, is what happens after the game," said Kompany, who recalled his own experiences of racism as a young player which included Real Betis fans "doing monkey chants" at him and former Anderlecht team-mate Cheick Tiote.
"After the game, you have the leader of an organisation, Jose Mourinho, who basically attacks the character of Vinicius Jr, by bringing in the type of celebration to discredit what Vinicius is doing in this moment.
"For me, in terms of leadership, it is a huge mistake. It is something we should not accept. I am very clear on that.
"The one thing you can't do is dismiss a person and attack the character of a person who's complaining about something he experienced and something that must be very painful to that person. There is something that needs to happen."
Kompany added: "I know 100 people who have worked with Jose Mourinho. I have never heard a person say anything bad about Jose.
"I understand he is fighting for his team and his club.
"I don't need to judge him as a person, but I know what I have heard and I understand maybe what he has done, but he has made a mistake."
Real Madrid boss Alvaro Arbeloa says Vinicius is "outraged" by the incident.
"Vini Jr. has been upset, as have we all. Above all, he has been very outraged by what happened. It's a racist act that has no place in sports or our society," he said.
"We have a tremendous opportunity not to let it pass and to continue fighting against this scourge that is racism. What's really important is to combat acts like the one we witnessed the other day.
"It's an intolerable situation that we don't want to see happen again. We will stand against it, especially when it involves a fellow professional. It's an act that must not and should not happen. Nothing, absolutely nothing, justifies a racist act."
The Brazilian Football Confederation has written to UEFA and FIFA stressing the need to "identify and punish" anyone guilty of racially abusing Vinicius.
It has been reported that the UEFA probe, headed up by an ethics and disciplinary investigator, could take up to three weeks to conclude.
