it is definitely a particular therapeutic 'technique' to deflect all personal questions. I think it has something to do with keeping the focus always on YOU. And it also maximizes any tendency you might have to project stuff (your own emotions, or seeing her as someone from your past) onto the blank screen she presents, which can be a very enlightening process if it becomes apparent. I have yet to ask my current T anything about herself, but I would like to know if she has kids too, since I have a lot of young parts who are not sure she even likes kids!
Anyhow, it seems like a reasonable thing to just ask her why she prefers not to answer questions. Maybe you could even phrase it in terms of what her therapeutic orientaion is, if you haven't had that conversation. But if course she might just ask back why you want to know! At which point you could tell her she's driving you nuts In any case you might then have a chance to process how you are affected...