Yes, we say this in Britain. It's mainly used by older people, myself included.

I know the phrase "dog's breakfast," not "dog's dinner." I'm bookish enough that I cannot say for sure whether I've actually heard it used in the United States, or whether I read it, possibly in a British book. In any case, the meaning is clear. Dogs vomit easily. (In the Bible, Proverbs 26:2, in traditional old English "King James Version, it says, Proverb 26:2, "As a dog returneth to his vomit, so a fool returneth to his folly." "Como perro que vuelve a su vómito...")

I have never heard it used in the US.