When there are 2 available birds of the same species the "merge" button will be lit up when you select the bird, and you will be able to push this. A circle will appear around the correct bird in your forest and you should select it and they will fly together and merge. To fully merge and level up a bird you will need to hatch the same bird 5 times - and merge them together, and feed them the required amount of Gnats until they are ready!
Once they have been fully merged you will get a maximum melodia harvest amount, and be on your way to completing an entire set.