I hate the water / electricity analogies because they don't make sense.

Voltage is not like water pressure. If you have a high water pressure, water will inexorably be pushed through the pipe to relieve the pressure. That's so regardless of whether the pipe is 1 inch or 1 yard in diameter.

Voltage on the other hand means nothing without current. A high voltage on its own - like static electricity - will not move many electrons through the conductor.

> Analogies illustrate a concept

Effectively illustrating the right concept is, in my opinion, the hardest part of communication. Analogies are powerful because they're intuitive illustrations.

> I typically see analogies used to support an argument, which analogies don't do

Yes and no. Analogies communicate concepts, and concepts support arguments. Analogies neither support arguments nor fail to support arguments; the concepts they communicate do that.

They're hard to use because they can be distracting: if you choose the wrong analogy, you may illustrate an irrelevant concept, rather than the one you mean to. That's why people argue: you mean to illustrate [concept A], but what pops into the receiver's mind is [concept A] AND [concept B]. So now you're not on the same page about what was just said.

Look at the cheerios example: how may related-but-not-the-same examples have popped up in the comments? It's a bad analogy: it needs to be much more narrow and specific.

They can certainly work as a proof, it's just one of the conditions on it being proof usually fail. Argument by metaphor says that A and B are isomorphic to one another and that since they are isomorphic, we can apply proofs from A to B (maybe with some modification). The problems come from either establishing the isomorphism or in mapping properties/predicates from one to the other.

argument by analogy is something like A -> B, C is similar to A in all ways that matter, thus (A -> B) -> (C -> D). analogies are powerful when the other person already agrees that (A -> B) and doesn't notice any important differences between A and C. if you use an analogy where the other person doesn't agree that A -> B in the first place, you'll never get anywhere. if they are really stubborn, they will come up with an endless list of reasons why A isn't quite like C, but at least you have a chance of refuting these.

analogies are not very good in arguments where the other person is resisting the conclusion you want to draw. they can be great when you are trying to teach/explain something to someone who trusts you though.

I think they can be helpful for achieving that initial critical mass of understanding around a topic ("a is to b as x is to y... oh okay, I better understand the relationship between a and b"), but yeah, if you don't unwind the analogy afterward, then it can be a false understanding.

I found this kind of thing a struggle in engineering math courses, where you'd often move equations into transformed spaces (frequency domain, whatever), perform operations on them, and then un-transform them to pop out a result. It's like, yes, the transform is obviously an immensely powerful abstraction, but I didn't really trust what was going on in there unless I did at least a few of the exercises from first principles as well, in order to prove to myself that doing operations in the transformed space was "safe".