Do you know why white pride is synonymous with hate whereas brown/black/gay/x pride is not? The latter movements were created to combat the systemic and social oppression of the other. They seek to undo the notion that white is the standard by which all others are judged. Imagine for just a second being bombarded from birth by media, authority figures, textbooks, classmates that you aren't as smart or pretty or capable as your white neighbors... those pride movements are meant to empower those marginalized groups from within against the long running history of white pride as a tool of oppression. White pride is not an internal movement. It historically seeks to maintain or gain power and influence through external means. It uses violence coercion and intimidation to assert the supremacy of whites over others. It seeks to maintain a culture of supremacy deeply rooted in social Darwinism and manifest destiny. It is a form of hierarchy.
As an anarchist, you are against hierarchy in all forms. That is one core tenant that all schools of anarchy share.
Why you shouldn't take it personally: white isn't a race. It is a social construct designed to reinforce hierarchy. If you are against all systems of oppression you have a real duty to examine it, even if it benefits you. That means combating the rational ignorance thats going to stop you from accepting that whiteness doesn't define you. It is your reaction to it that will say whether or not you are dedicated to equality, real liberty and combating oppression.
Lastly of none of this makes it through to you, ask yourself what part of your culture can be defined as white? If you're Irish or Serbian for example and want to appreciate your history, is that defined as whiteness? I would say no, those cultures predate the notion of whiteness (as was further explained in the above link). If you cannot separate them, its going to take some introspection to find out why... and that doesn't invalidate the position of somebody who disagrees with you simply because they can't (or don't want to) explain it to you in a way that slips past your defense mechanisms.