Birds are mentioned several times here, and other things are compared to them; so are fish. The speaker repeatedly turns to animals in order to define his own situation. Birds can fly anywhere they want in the vast sky and fish can swim just about anywhere they want in the vast ocean. These animals enjoy a lot more freedom than us land-locked folk. The speaker, however, claims that this isn't what freedom is all about, or rather that one doesn't need to be able to fly and swim underwater to be truly free.
Lines 7-8: The speaker mentions "birds that wanton in the air." The birds are a symbol of freedom that, for all their flying, still possess less freedom than the speaker.
Lines 15-16: Not even the fish that "tipple" in the "deep" enjoy the same liberty as the speaker and his friends. These fish symbolize wild animals that enjoy a whole lot of freedom.
Line 17: Now the speaker compares himself to singing "linnets." Since he uses the word "like," we know we've got a simile on our hands.