If you read about how Claudia sees her home life at the beginning of From the Mixed-Up Files of Mrs. Basil E. Frankweiler, you'd think that she was stuck in a prison instead of just living a normal childhood life. But when she and Jamie break free, they end up going to a place where they are often locked in and then set free. At the museum, they have to hide and stay inside when the museum is closed, and they can only leave during the day. But they're confined by their own decision—they want to stay at the museum, after all. So in a way, by staying locked up at the museum, Claudia and Jamie are really exercising their freedom. How about that?
At the museum, Claudia and Jamie find the freedom to live the kind of life they want to, regardless of their family, expectations, or age.
Claudia relates so much to the angel statue because she feels trapped or confined in the same way it is.