But that's the thing. She never said "people are trying to take that away from me." She said "You can't take that away from me." So you and she are in agreement in that one very narrow area. No one can take her lived experience away from her. That is, I believe, what she's saying.

Looking at the context seishonagon gives above: Sarah Palin was asked to avoid talking about (among other things) the military. My interpretation is that Palin's response means: "My life experiences involving the military (through my son) are an essential and important part of how I define myself and who I am as a person, so asking me not to talk about the military is unreasonable and also kind of insulting to me and my son. My life experiences involving the military are an essential part of me, and I can't separate myself from that." With a possible subtext of: "And we all know there are those people out there who don't support and respect the military the way they should, but you, the people I'm talking to, are not like that, so we're all part of the same group and should bond together against unspecified threats."

That's how my mental conservative-parser reads it, anyway. I could be wrong; I'm not inside her head. (I'm trying to avoid expressing any personal opinions on the content of her speech here, because I think that's irrelevant to the subject. The only thesis that I'm trying to support is that her comment had an understandable meaning.) Or maybe I'm over-analyzing now and "they're trying to take X experience away from me" is just shorthand for "they're trying to take the validation of X experience away from me".

As others have said, she wasn't speaking very precisely. It sounds like she made an emotional utterance on the spur of the moment and got an emotional response from her audience, so I'm not sure how much more analysis one can do.

I can see the analogy to being queer, which it is is completely valid to say is also an inseparable part of one's life experience and identity.