Since the late 19th century, the expression dog's breakfast has been chiefly used in British slang for a distasteful, inedible mess or mixture, literally and figuratively.
The bill is a dog's breakfast of legal errors and technical impossibilities.
— Larry Downes, Forbes, 24 May 2018
A couple etymologies on the expression have been served up: it could be an allusion to a culinary effort that results in something that is only fit for the consumption of one's four-legged companion or the fact that dogs' meals in the past (before canned and packaged gourmet dog food) were a mess of leftover "people" food. But, honestly, the connection between a dog and breakfast isn't at all clear—nor is dinner, for that matter, which is the next course we serve—since a dog's breakfast is usually the same as its lunch and dinner.
The term dog's dinner fashionably arrives by the beginning of the 20th century. Like dog's breakfast, it refers to a figurative mess but additionally connotes that someone or something is a mess concerning their dress or appearance.
A survey a couple of years ago … revealed that ... Brits love a good project in the home, … with one in 10 of us actually injuring ourselves doing DIY, and there being on average more than 31 million unfinished jobs in British homes started by amateur improvers but never completed. What's more, we spend on average almost £250 getting an expert in to fix what we've made a dog's dinner of.
— David Barnett, The Independent, 15 July 2018
It's unclear how dog's dinner became associated with one's appearance. Another expression linking the two is "to put on the dog," meaning "to pretend that one is very stylish or rich"—but it's American and, as such, is unlikely to be cut from the same cloth.
We always had a Saturday night party when Aunt Lizzie came. That gave her a chance to visit those she had grown up with, and as Father said, to show off, and put on the dog, and generally to show how well she had done since she left Renfrew County.
— Mary Cook, The Smiths Falls Record News, 18 May 2018
Dog by itself, in reference to affected stylishness, nuzzled its way into American English in the second half of the 19th century. The following excerpt from the 1871 work Four Years at Yale offers confirmation: “Dog [means] style, splurge. To put on dog is to make a flashy display, to cut a swell."