How do you explain the "祖" in "祖国"? As you know, "藉贯”means the birthplace of oneself but "祖藉" means the place of living of one's ancestors, so when someone says “我的祖国”, does he mean the country of his ancestors or the country of his own (for example, he is an American Chinese born and living in the US)?

The latest edition of 新华字典 says that 祖国 is the country of one's own(对自己国家的亲切称呼）（Page 674,新华字典第11版，商务印书馆，2012）, so it is very confusing, what does 祖 here mean?
祖 generally means ancestor, 祖国 means someone's own country and MAY have the implication that his/her ancestors lived in the same country as well, although it doesn't always have to be the case. I think 祖国 is similar to "motherland" in English, an expression to imply close relation. As an analogy, although "mother" and "land" give you a feeling what the word "motherland" means, being really serious about the meaning of "mother" itself won't help much here.

P.S. As a Chinese, I usually perceive Chinese words as a whole rather than separating every character and studying them, but I'm not sure if this is the best way for learning Chinese as a foreign language.