Yes, that's right: it's a form of physical punishment. There are two aspects to it:
- A verbal transgression such as lying or swearing makes the mouth (metaphorically) dirty. Soap (metaphorically) cleans it out.
- Metaphor aside, it's rather unpleasant, so it's an effective punishment.
This sort of punishment isn't very common anymore, but people often refer to it in speech. One cliché response to profanity is "I ought to wash your mouth out with soap!" And we also say people have "dirty mouths" when they use profanity.
This punishment was and is real, though. He really could have gotten a bar of soap in his mouth as punishment.
Thanks for supplying the context. You're correct in your understanding of "over his milk".
The oxcart question is not offensive in itself, though it is colloquial and hence sounds a bit silly.
The key sentence in the larger context you provide is "She had, for almost two years now, baited him. In the halls, in the elevator, even in his own office, into which she romped now and then like a circus horse, she was constantly shouting these silly questions at him."
To bait someone is to deliberately taunt or annoy them, hoping to get a response from them. (Analogous to offering bait to a fish in the hope that it would bite.) She behaves extravagantly ("like a circus horse"), she shouts out rhetorical questions that are both colloquial and hackneyed, and ultimately she is always asking the same question that assumes things are going badly. She's negative, imposing, and annoying.
At least that's my take on it, based on a few paragraphs. I have to admit that I've never read that classic story before.
Best Answer
The first paragraph gives an insight about this.
It's probably a tongue-in-cheek comment by RNC spokeswoman Kirsten Kukowski. Check out the meaning of the words used here:
In this context, it means that the president simply did both - political work by addressing the 'water shortage issue' but (then) also playing golf on the ground that requires 'too much water' to maintain.