I had this situation at work where a colleague (not my report, but junior to me by 5~10 years) was pushing people so strongly to work on a project that it caused a few of those people and their managers to come to me and either complain outright about it or just express confusion about it, because they hadn't heard about who this junior person was.
I felt like I owed this colleague to share the feedback that I was seeing a pattern that could be detrimental to their career and shared it, specifically pointing at instances and offering suggestions.
Here's the problem. I'm a man, that colleague is a woman, and I work in an industry (as many) where women have reason to complain of multiple biases. I was very conscious that my feedback could come off as sexist, so if anything, I held back until I saw more of a pattern, and made a very conscious effort to focus on the specifics without using any adjectives to describe her personally -- I said things like "in this case, you did this, and the outcome was that, here's how it could become a problem for you in the long term, and here's a couple things you can consider doing in the future", nothing at all like "you're being too aggressive". She thanked me for that feedback, shared some of her own, and I thought that feedback session went well.
Still, a couple days after sharing this, she told me that the conversation made her feel like I was basically just a man telling a woman to shut up.
I felt very sad hearing that because in reality I am completely rooting for her and want her to succeed, and her questions are making it very uncomfortable, if not dangerous career-wise, for me to help her by sharing insights.
I pointed out that I had seen a pattern not a single instance, that I had "checked my biases", that in fact a few of the people complaining were women too, and that she could do whatever she wanted with that feedback. I don't know if she was genuinely convinced but we left it at that. Not a pleasant experience.
So here's my question. What more can I do to give feedback to someone and completely avoid giving the false impression that my feedback comes from prejudice? (And replace gender by race/ethnicity/age/religion/sexual orientation etc...)
A couple clarifications addressing some of the things in the comments or replies.
I had discussed the situation with this person's manager, who had encouraged me to share feedback and help coach her rather than he having to do it with secondhand information.
The person didn't argue that gender bias was a reason to dismiss my feedback. She acknowledged the feedback I gave her, just said it made her feel a certain way, which both of us disliked.
For my first question, I am overwhelmed with gratitude at this community for the amount of thoughtful discussion. Thank you all.
I also want to explain the answer that I'll accept, because the topic is obviously contentious, answers are a bit all over the place and because the one that I've personally found the most useful is by far not the one with the highest "popular vote". I'll go over the most popular answers and my thoughts on them first.
Old Nick's answer is relevant, but not a direct answer to my question; its main argument is that I shouldn't give feedback directly to a person who does not report to me. Even if that were true, I will be confronted to situations where dodging the responsibility of giving feedback is not an option, and then it won't help me.
ShinEmperor's answer construes the situation as one where I have given "advice" to the person, and then argues against giving advice in general. But there is a big difference between advice (where I have nothing at stake in the other person's behavior) and feedback (where I do). This situation had affected me because it destabilized my team, so doing nothing also had a cost to me; I had skin in the game to defend.
I generally agree with bilbo_pingouin and Words_like_jared's answers, but instead of giving me thoughts on what I could have done differently, they focus on what I should do from now on.
So in the end, it was in Erika's answer that I have found the most interesting clue about something I could have done differently, which is to ask questions in the first place. I was certainly focused on giving feedback, and I can see how asking questions instead might have helped in this situation. The answer is slightly off-base in other respects due to the vagueness of the context in my question (i.e. the answer makes it sound like it was someone pushing a pet project, which it wasn't) but I'll accept it as the most useful to me personally.
Again thanks everyone for the thoughtful discussion and answers.