Hm...well, the most common blue bird is the...well...
bluebird. LMAO
However, in reference to the song, Ayumi is most likely referring to the "blue bird" as, like the others have said before, her lover. She's telling him, "Let's go together, across the blue sky" so she probably means that she wants to be free with him forever. So pretty...reading the translation made me realize what a good lyricist she can be.