"Milkshake Duck" is an internet slang term for something amusing or wholesome on the internet that quickly loses its lustre. Its the type of slang term that you didn't know you needed until someone invented it.
In the case of "Milkshake Duck", the inventer of the term was Twitter user Pixelatedboat, who on June 12th, 2016, tweeted out:
The whole internet loves Milkshake Duck, a lovely duck that drinks Milkshakes. *5 seconds later* We regret to inform you the duck is racist.
The joke being, that in a world where the internet has been recording our random thoughts for the past 20 years, and where morals and standards have changed, pretty much every content creator, no matter how seemingly innocent, probably has some type of skeleton in the closet. "Milkshake Duck" is the feeling when someone watches a youtube video where a pleasantly goofy young man plays Beatles songs on the ukulele, and then discovers that the singer also has a number of videos explaining how the Jewish media disguises the fact that the world is flat.