Bosom Friends by Angela Brazil

Two girls meet at a seaside town that they are visiting.  One's name is Isobel Stewart, and the other's name is Isabelle Stuart, and they think that the similarity of the names mean that they should share everything together. Both of the girls join in the fun with a group of other visiting children, though Isobel enjoys the sports and adventures more than Belle, who prefers dressing up and attending parties. As the summer progresses, the two of them find that, although they are close friends, things are not as cheerful with their friendship as they might think.