So I came across this polyethylene bottle half-full of what I guess may be KOH. What is a good way to confirm what is in there, as opposed to sodium hydroxide, or who knows what?
Why might you be guessing this?
Because I can make out some faded magic marker reading “K[illegible]H”
I think the most distinguishing characteristic is ‘causes burning on skin’ - caustic, that’s the word.
If you can safely perform a flame test, you should see the prominent ~590 nm (yellow/orange) D line for sodium vs violet for potassium.
ETA:
They’re both caustic.
Is it from a soap factory maybe?
That sounds like something relatively easy to try. The material itself is in the form of prills. I also have some (known for sure to be) sodium hydroxide.
It is from some guy who apparently had it with a bunch of stuff for many years, and I do not know where it originally came from nor am able to ask the guy.
[Now that you mention it, once upon a time I did score some bottles of glycerin and other stuff that were discarded from a soap factory!]
There are also a couple of brown glass bottles of… could be anything at all, really. Mystery acid? Solvent? 
Yeah, a flame test is the easiest method to determine if it’s potassium versus sodium. Dip a clean wooden stick, or ideally, some platinum or nichrome wire in the solution then hold over a flame. If you really want to be formal, clean the wire with HCl first, but if you don’t have any, holding it in the flame for a while first is probably good enough for this test. (Oops, I just saw that you said it’s in solid form. So yes, you can just wave a flame on it, but safety is a bigger concern.)
I don’t know any of any other relatively common chemicals with the formula K_H, other than potassium hydride (KH) which can be pretty dangerous. I suspect it would have already blown up if it was actually KH.
A comforting thought for sure as the OP is playing a flame over a sample of his mystery product. 
The science and practice of chemistry just demands a certain dry humor when it comes to lethal hazards.
Out of curiosity, how will the answer affect what you do next? Do you have an idea of what you will do if it is (or isn’t) KOH?
Not the OP, but sodium soaps are firmer, whereas potassium soaps are soft or liquid.
Since the bottle just says K(something)H, you could just start by dissolving a bit in water and testing the pH. If it’s soluble and very basic, that’s a point in favor of KOH.
For distinguishing between NaOH and KOH specifically, there’s a few things you could do. As mentioned upthread, flame test would be easy if you can do it safely. If not, KOH is much more soluble in methanol and ethanol than NaOH (like, twice as soluble), and slightly more soluble in water. You could also do a titration if you have the equipment; dissolve a known mass in water, then titrate with a strong acid until neutral. You then know the moles of acid required, and can work out the molar mass of the base.
I anticipated this question 
As it happens, I do not at present have an application in mind for which I need potassium hydroxide (and I usually prefer hard soaps), but the sight of the unlabeled bottle is nagging at me, and I figure it will not hurt to verify and know for sure what are the contents.
Sounds like good chemical hygiene to me; carry on. 
I would be happy with that, since potassium hydroxide is labeled KOH while sodium hydroxide is labeled NaOH. It’s not exactly a Wheel of Fortune task.
I use KOH routinely in my tanning business, it works appreciably better than NaOH in this application, and I can tell if a hide’s been swelled in a NaOH solution instead of a KOH solution, but I don’t think the tactile differences are very cross-applicable.
Both substances absorbing water and reacting with CO2 makes this less precise. But the pellets not having turned into deliquescent goo suggests a good seal.
Optimism is very important in a chemist working with unknown substances!
Factual answer to factual question - good ways include:
- A degree in chemistry and a modern, well equipped laboratory.
- In particular, any of several mass spectroscopy or liquid chromatography methods.
- As a fallback, the host of methods invented by chemists for exactly this kind of question
- Paying someone else to do the above
Why do you assume that the scratched up label on the bottle bears any relation to the contents?
Any of the methods mentioned by other poster are likely to give some result. In some cases, this result will be published in the newspapers. In a few cases, the result will even be mentioned on television.
One guy (I am pretty certain not a professional chemist, though) mentioned a possible concern about mercury that could be released by burning the substance, if it was produced using a mercury cell process. (I subsequently found a document that in a 1995 study average mercury levels in these hydroxides were found to be 0.1 part per million.)