Translation:The colors of the shoe are violet and pink.
Why is "The color of the shoe is violet and pink" wrong? How do I know if its talking about a pair of shoes rather than just one shoe?
That's not the problem. In the English sentence, you have multiple colours (plural), but you only wrote "colour" as a singular noun rather than "colours".
But to answer your second question, to know the shoes are plural, there'd be a word indicating plurality, like "các", "những", "đôi" (pair), or even a number.
No, his English sentence is what the vast majority of native English speakers would say. If something, like clothing, has multiple colors, we still talk about it in the singular. We would say something like "The color of the shirt IS black and white". We probably wouldn't say "*The colors of the shirt are black and white" (note: the second sentence isn't wrong, but more English speakers would prefer the first)
I hate this question, I keep getting it wrong, repeatedly, and it's not because I don't understand the Vietnamese meaning, it's because I am forced to put down for the 'right answer' something that sounds very unnatural to me in English and other more natural-sounding answers get rejected. That should not be so.