"Why" is sometimes used as an interjection or exclamation to express mild surprise, hesitation, approval, disapproval, or impatience. For example:
- Why, that useless man! (disapproval)
- Why, I was just thinking about you when you called! (surprise)
"Why" isn't the only interrogative word to be used this way, for example "what luck!"
In the context of your example, it would seem to be expressing surprise. However, why it is used twice in your example I have no idea. It seems wrong. Perhaps you have misquoted it, or there is some missing punctuation? If a person was especially surprised, I suppose they might stammer and say it twice?
Why... why I have two oak trees in my backyard too!
Although I can't see what is so surprising about having the same trees.