For all their delicateness and nurturing instincts, birds are also resilient and migrate to wherever conditions are better for them. And, in situations that are dangerous to them, they will abandon their nests since the survival instinct supercedes all others. Indeed, the use of the bird as symbolic of Nora is appropriate.
Nora is seen as an entertainer and comfort for Torvald, like a pet.
Birds also are fragile creatures with hollow bones. They sing and are seen as merry, but they are also seen to need protection from harm.
The household is also like a nest, a place to be "feathered" with comforts, in Torvald's view. Nora is the bird to do the feathering and give him comfort.
Nora sings like a bird as if she is happy. However, she is not happy. Like a bird which is kept as a prize in a cage, Torvald keeps his little songbird, Nora, in a cage. She has no freedoms. She has many restrictions on her life because she is a woman. Women in the nineteenth century were idolized but not respected, as the above post states. Torvald idolizes Nora, but he does not respect her. He calls her featherbrain and scatterbrain. These are not affectionate terms. They are insults.
Birds can also be symbolic of beautiful, delicate creatures. Women are often compared to birds because they are considered fragile. Birds are normally appreciated by most people, but they still lock them in cages. So while they are appreciated, they are not respected.
We generally associate birds with flight and freedom, but Nora is like a pretty pet bird that is kept in her cage, her doll house. The bird may be alive and well in the cage, but something essential in her nature is thwarted. In the case of Nora, societal expectations have clipped her wings. The conflict of the story is whether Nora will break free of those restrictions and fly on her own, or stay within the safety of her life as it is.
But couldn't Nora be restricted by the enviorment she lives in? Women in the 19th century didn't have freedom and were treated like an amusement tool!