It was some time in 1992 when I spoke with Maxim on whether the old (i.e., large) sizes would still work. I was told to use the 1uF to avoid harming the chip. I have done so ever since.
They undoubtedly went to a higher switching frequency so they could reduce the size of the capacitor to make the product more attractive. If you ask, they probably give you the conservative side anytime the person you're talking to is not totally sure. Looking at other switched-capacitor voltage converter datasheets, I don't find any upper limit to the capacitor size, except that they say that going beyond a certain value for a given switching frequency won't give any benefit. The story, as I remember it from one of the industry trade magazines, is that the MAX232 spawned a gob of switched-capacitor voltage converters because it was found that a lot of customers were using it not for RS-232 but for power conversion. I don't know when the 7660 came along, but there are a lot of variations on that, some of which I've used (from Linear Technologies, National, Intersil, and Maxim), variations for higher current, higher voltage, even regulated, etc..
The reasoning behind using tantalums instead of electrolytics is the former exhibits a lot less leakage, thus making the charge pump more efficient. Electrolytics will work, just not as well.
Leakage of electrolytics is quite minimal. I've kept memory alive on a 10uF for a day (didn't try any longer), and a product we had 20+ years ago charged a 220uF 'lytic through a 1M resistor to get a 220-second time constant, and it was consistently and surprisingly pretty accurate, even without using precision or low-leakage capacitors, although it was at temperatures that were comfortable to humans (ie, we didn't have to run it hot).
Tantalums will have lower ESR. If you can get OS-CON capacitors, that's even better. OS-CON is not a brand but an electrolyte type, and there are at least two companies making OS-CON capacitors, one being
Sanyo (which we use for the small switching supplies in our products).
I also got a new multimeter which has a frequency counter (I am hoping I can see the clock signals with it).
What's the upper limit? My DMM has an upper limit of 200kHz on the frequency counter, and I can hardly imagine connecting the long leads to a MHz or higher square wave. (I'm not saying it can't work for that, but I'll be interested.) After I wrote this, I see you added:
Don't know if this meter is able to measure square wave frequency or whether it is AC only - Manual doesn't say.
I expect it is AC-coupled, so the DC component shouldn't matter as long as it's not so high that it blows the input capacitor (like a few hundred volts).
This will show logic High / Low and pulsing (ie clock). I've yet to understand what this gives that a multimeter doesn't, apart from knowing a line is pulsing
A 0-5V square wave with a 50% duty cycle would probably just appear as 2.5V on a multimeter, as if it were sitting there quietly as a DC voltage. Now you'll know it's actually pulsing between two valid logic states.