In 'BioShock Infinite', Songbird is an intriguing character. It's an imposing creature, a creation of scientific genius, bonded with Elizabeth. Despite its intimidating stature, it's shown to have a unique and deep bond with Elizabeth.
However, the game doesn't offer an option to romance Songbird. The connection between Songbird and Elizabeth is more about companionship and less about romantic engagement. There's a whole lot of interesting backstories about Songbird that could fill a novel, but unfortunately, romantic interaction isn't part of it.