From your question, it seems that you do understand this quote. Anzaldua is saying that even though there are aspects of her culture that she hates and even though she is strong enough to leave her home and culture, there are still parts of her that always remain Mexican.
I cannot think that it would be possible to argue against this statement. I believe that it is impossible to separate ourselves from the culture we grew up in.
I have seen this in my father who left the Philippines 49 years ago now but has elements of his native culture still engrained in him. I can see it in myself since I grew up outside the US and still feel that I am different from other Americans because I grew up in a different culture.
Even if we reject most of our birth culture (or if it simply fades as time goes by) it is still where the foundations of our personalities were built and that is something we can never completely lose.