I'm sure situations like you describe do happen but it's been my experience that most kids go full tilt one way or the other.
My experience has included a full range of middle ground with full tilt being less common.
My daughter never cared for dolls. She liked some "girlie" stuff, but has always preferred swords, nunchucks, dragons and fighting. Given the option for gymnastics or martial arts, she has excelled at the martial arts. Yet her appearance, by her own choice, is very girlie. As puberty is beginning (TOO SOON!) I see certain changes in her interests, but she still dislikes pink and all things like that.
My son started playing with trucks before he knew what a truck was. He seemed to be going the "full tilt" way. Except for those times he sneaks into his sister's room, tries on perfume and says "I smell beautiful", or plays in some of her clothes.
Being a homeschooler I spend a lot of time with other people's children. Especially in secular families, I see less of the roles being enforced (this is for boys, this is for girls), and a lot of middle ground. Most of the girls don't go the doll route. Many of the boys play with dishes, pots and pans, kitchen toys generally considered "girls" stuff. Oddly enough, it's the very religious families where I find the doll collecting and more strictly imposed rules of gender. But it's anecdotal, who knows really?
As for the other comments, I simply don't have enough information about this family to know how I feel about the situation. As a parent I've never faced it. While I always knew I'd accept my child and love them exactly the same if they were gay, straight or inbetween, this kind of issue coming up at such a young age really threw me.
The two people I know personally who have changed their sex have serious issues that didn't appear to be solved by the operation. The first person started off as a straight male. He had a much older sister who grew up to become an NFL cheerleader. She was always living in the shadow of her sister, and it was glaringly obvious a lot of major baggage came with growing up in this environment. The gender reassignment compounded the issues in a lot of ways, and she seems to struggle with life even more than before. I don't discount the experiences of other transgendered people who may not have had environmental factors involved, and certainly my limited anecdotal experience cannot speak for more than these two people.
I can't possibly know what it's like to find oneself in the situation where they believe strongly they are living in the wrong body in such a fundamental way. If this is truly the case with this child, ultimately I'm happy he/she has parents who will be supportive and accepting. I'd feel better if the child had psychiatric care, that's all.
I wouldn't discount puberty has having the power to make a profound difference. It could very well make no difference at all, but it could.