A "doggy bag" is when you don't finish your meal, often heard in relation to a Chinese restaurant, and get the rest as a takeout. I don't know where that name came from or what it has to do with dogs, but to me I slightly cringe every time I hear it, not just because in China eating dogs is not a taboo, but because where I come from there is only one purpose that bags have in connection to dogs, and it's of a ... very different flavor, heh.
Apart from that I also find it weird how humans' food consumption is in any way associated to dogs. If I didn't know this colloquial but common term, I'd feel disrespected if someone offered me to put my food in a doggy bag. Am I supposed to be a good boy and eat up then? Or are they suggesting I eat like a dog? Not worthy of a plate?
Then again, this works both ways. I'm not sure, but English speakers might react in a similar way when they hear Germans call their cell phone "handy".