When I was attending a predominantly Native inner-city school for grade 7, a girl told me I was "too white to be a person".
To their credit, the teachers took that seriously, despite ignoring the many other hurtful taunts I got on a daily basis because I was a nerdy undiagnosed autistic kid with mental health issues. They challenged her about how she'd feel if someone said the reverse to her.
Unfortunately, many people wouldn't take that nearly so seriously. There are a lot of people claiming that 'reverse discrimination doesn't exist'. This article
specifically lists off reverse racism, thin-shaming, and misandry as types of reverse discrimination that "don't exist", and claimed that they don't cause systemic harm.
My own brush with reverse racism didn't cause harm because it happened once and received an appropriate response. If I'd been told "stop whining, white people are privileged so it's not a big deal to make fun of them", then it would have caused harm, just as all the other times my experiences of bullying were invalidated. And the girl who'd insulted me would have learnt that it was OK to make hurtful comments about my skin color. So it might not have been such an isolated incident.
And especially misandry. I'm not a man, but even I can see that both sexes are targeted by sexism. My WGST prof outright claimed that men couldn't be raped because an erection indicates consent, and accused a music video depicting a male musician's experience with an emotionally abusive woman partner as sexist because it showed a woman in a negative light.
She's not the only one. The idea that "women can't victimize men" is something male victims of female-perpetrated sexual assault and domestic violence run into on a regular basis. Even male child sexual abuse victims are less likely to be seen as victims, especially adolescent victims of sexually abusive women. (A 25 year old having sex with a 13 year old is seen as assault if it's a man and a girl, and as a "kid getting lucky" if it's a woman and a boy.) The harm of not taking victimization seriously is very severe.
In addition, fathers are subject to prejudice when involved in active caregiving. Men's bathrooms rarely have changing tables, paternity leave is harder to get and brings more negative judgment than maternity leave, a man sitting at a playground watching children is assumed to be a pedophile rather than a father, and men are told they're "babysitting" their own children. All of these enforce the idea that men aren't supposed to be active fathers. Some men are distanced from their children, harming both them and their kids, while those who stick it out face consequences for doing so.
As for thin-shaming, body image is a problem for women of all sizes. Even thin women frequently feel ashamed of their bodies. And for men, thin, nonmuscular men are seen as weakly and "feminine", definitely not the ideal. This harms men by encouraging them to endanger their health to build more muscle, exercising in dangerous ways and in some cases taking steroids.
The discrimination is often less pervasive and has a different flavor to it, but the same can be said when you compare the experiences of Jews to blacks. Similarly, my experience as a sexual abuse victim with supportive parents who immediately acted to protect me on their first suspicion of abuse and stood up for me when others tried to invalidate my experiences is very much different from the experiences my abusers had, where their abusers included at least two different parental figures, and no one acted to support them until adolescence. Their abuse experiences were far worse than mine on pretty much any dimension that you could measure abuse severity. But my abuse experiences were still wrong and hurtful.
There are also examples of historical situations where the discrimination has reversed. For a period of time in England, religious discrimination regularly reversed depending on the beliefs of the current monarch - Catholic monarchs persecuted Protestants, and Protestant monarchs persecuted Catholics. A number of colonized countries, during revolutions, reversed the racism in their society and began persecuting whites.
Some people do make false claims of reverse discrimination, but again, that's true in other cases, such as the girl my brother knew who claimed a boy was "raping" her if he accidentally bumped into her, or the Jewish classmate he had who claimed anyone who objected to anything he did was anti-Jewish. Or the boy who accused me and my friend of not wanting to play with him because he was disabled, when in reality we just didn't want to play with a stranger.
It's important to recognize the historical weight behind some forms of discrimination, and how it hasn't been supporting other forms of discrimination. But discrimination is still wrong, even if it's directed towards someone with historical privilege. A double standard that judges the same act differently depending on the identity of the target is wrong, regardless of the historical benefits or disadvantages of the group. And if one person has suffered 3 wrongs, those are no less wrong just because someone else has suffered a thousand.