How do I get Feathers on my iPhone keyboard?

The short answer is: you can't.

The symbol keyboard and the styles picker can only be used within the Feathers app.

The longer answer is: I wish you could but it's just not possible for me to implement since, unfortunately, Apple does not provide a way for 3rd-party developers to add to the system keyboard.

What about the Emoji keyboard?

The Emoji keyboard is a keyboard built by Apple that is active by default on Japanese iPhones. The Emoji apps simply activate it on iPhones outside Japan too – they don't add a new, non-Apple keyboard to the system keyboard.