The symbol of a bird in a cage is a classic (and sometimes overused) analogy for a loss of freedom. It is in the bird's nature to fly, and yet flying is impossible inside the cage. Thus we might consider living in a cage to be completely antithetical and antagonistic to the bird's nature, its "purpose" in life. This is the perspective by which the bird is interpreted in Siddhartha. We might go a step further and imply that the bird is everyone; anyone can attain freedom and enlightenment, but the details of our materialistic lives trap us.
Siddhartha's dream, in which the bird has died and he throws its corpse out the window, seems to send the message that death and/or suicide is not an appropriate resolution to Siddhartha's problems; that this would be the same as "throwing away all that was good and of value in himself". This may represent a maturing of his worldview beyond the need for escape by any means necessary.
Kamala sets the bird free in a way which coincides with Siddhartha's spiritual enlightenment, completing the analogy and equating the bird's freedom with Siddhartha's maturation beyond the things that have been making him unhappy. This also symbolizes that she is "letting him go", and demonstrates generosity on her part because this act cost her something she valued, i.e. her relationship with the thing being set free.
You have put your finger on one of my very favorite parts (and favorite symbols) in Siddhartha. You are correct to suspect that this cage has amazing symbolic significance. The songbird represents Siddhartha while the cage represents his life (at this point) that traps him. Let's learn why while looking at the text.
The songbird in question rests in a cage for the entire day while he sings and sings and sings (apparently wishing to be set free). Siddhartha has a revelation while looking at this songbird whom Siddhartha both feeds and watches. You see, Siddhartha is in a cage of his own: his current place in life. Just like the songbird can't be happy, Siddhartha can't be happy either. Siddhartha finds himself wishing for the death of the songbird out of compassion. In fact, he has a dream that this songbird of Kamala's dies. Siddhartha promptly takes the bird out and throws it on the road, but he immediately feels regretful. This makes Siddhartha realize that he is oppressed in his current life station and needs to escape somehow. Siddhartha realizes that obtaining money and material wealth doesn't lead to happiness. Siddhartha realizes that he is not able to truly love another person, so he can't find happiness that way, either. This leads Siddhartha to wish for escape from his current life in order to find his innocence again.
For much longer, he could have stayed with Kamaswami, made money, wasted money, filled his stomach, and let his soul die of thirst; for much longer he could have lived in this soft, well upholstered hell,...
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if this had not happened: the moment of complete hopelessness and despair, that most extreme moment, when he hang over the rushing waters and was ready to destroy himself. That he had felt this despair, this deep disgust, and that he had not succumbed to it, that the bird, the joyful source and voice in him was still alive after all, this was why he felt joy, this was why he laughed, this was why his face was smiling brightly under his hair which had turned gray.
In conclusion, let's talk about Kamala. In regards to Kamala, even though the songbird still symbolizes a trapped Siddhartha, she takes a different approach to freedom. Where Siddhartha wishes for the bird's death for it to obtain freedom. Kamala realizes that she can be part of the process by setting the bird free. Kamala sets Siddhartha free as well. He needs to follow a different path and will not return.