How does "migrating birds have lost the south" fit into this song?
well as you can see at 如果两个人的天堂, the lyric tell us that there are a couple, 像是温馨的墙  tell us that the man/girl feel that their love like just like a wall, 囚禁你的梦想 imprison the his girlfriend / her boyfriend dream, 幸福是否像是一扇铁窗 then he begin to think, is this happiness has become a barred window that imprison the dream, 候鸟失去了南方 just like group of bird need/want to migrate to south, but they can't.

It's a very interesting metaphore, bird are usually migrate to the south when north become cold, it is something they need to do. In this song the writer illustrate that dream is something that as important as bird need to migrate. But when we has somebody that we love, somehow that love is stronger than the dream that we need to pursue. So the writer say this love just like prison that imprison us from pursue our dream.

Hope that explain your question, this is a very good song, I heard it once years ago, and what a surprise i can hear it again when i open your question.