With chips like that it's best to follow the manufacturer's design closely, unless you really know what you are doing. Ceramics are generally preferred in that sort of application, they are smaller and more reliable than electrolytics, handle high temperatures better, and often have a lower ESR.
ceramic should work as long as you meet the requirements in the datasheet: 0.1ohm < esr < 5ohm and srf > 1mhz.
Its probably easier to find those properties in a tantalum cap, especially back in 2002 when that datasheet was released.
EDIT: Some more info about LDO stability and why the ESR has to fall in a particular range.
A generic LDO works by comparing the output voltage to an internal voltage reference with an error amplifier and driving a PNP transistor to correct for this error.
The problem comes in when you look at the phase shift and loop gain of this feedback path. The error amplifier and the load being driven both contribute poles to the frequency response of the feedback loop. These poles act as a low pass filter resulting in loop gain decreasing as frequency increases. As we know a pole also introduces a negative phase shift. If this phase shift is allowed to reach -180deg the feedback loop becomes unstable and the LDO will oscillate.
What this means is that every time the error amp tries to compensate for an error the result of its correction is 180deg out of phase, or inverted, consequently the error amp is basically thrown for a loop and begins making the opposite correction that it should be making, resulting in wild instability.
To avoid this situation we need to prevent the phase shift in the feedback loop from ever getting to -180deg, actually we only need to keep it from reaching -180deg within the region that the LDO can generate gain > 1 as the damped response of the system past this point will prevent oscillation. This frequency is defined by the unity-gain point of PNP pass transistor.
The way we prevent this phase shift is by using a capacitor with a ESR in a certain region. The capacitance will shift the pole created by the load but more importantly the ESR will contribute a higher frequency zero. Basically you've added a high pass filter to the feedback loop. The phase shift introduced by the ESR will work to counteract the phase shift introduced at lower frequencies by the poles from the error amp and the load.
The reason that the ESR has to be in a particular range is that if its too low, the zero contributed to the frequency response will be located very high in frequency, above the unity-gain point of the pass transistor. As a result its not effective in making sure the phase shift of the feedback loop doesn't reach -180deg before the unity-gain frequency.
If the ESR is too high, the zero will be very low in frequency. There is another pole in the frequency response created by the parasitics of the pass transistor, if the zero from the capacitor ESR is too low in frequency, this pole will be reached while we still have gain > 1, this will cancel out the effect of the ESR zero and we will likely reach -180deg phase shift before we reach unity gain.
All that said, these problems are indicative of older LDO designs. Many/Most/All new designs include additional internal compensation in the feedback loop which uncouples LDO stability from the ESR specification of the output capacitors.
Best Answer
In order to design a stable regulator, the nature of the output load must be understood. In the old days, when ceramic caps were harder to come by in large capacitances, most regulators relied on tantalum or aluminum electrolytic caps for output filters. These have a moderately high ESR. Accordingly, the regulators were designed to be stable with relatively high ESR output caps. Low ESR caps might not be stable in such old regulators.
When tantalum caps were very difficult to find in the late 90's, a lot of IC makers designed their newer regulators to work with ceramic caps (which have low ESR at high frequencies but not necessarily at line frequencies). Often this was a selling point and may even be mentioned in the datasheet. Over time this became the norm, and still is today. Ceramics are cheap and easy. The only real drawback to ceramic is that if they experience shock, they may produce voltage spikes which can lead to noise. They can be mechanically brittle, also. But very often they are a great, cheap and easy choice.
You can mimic a tantalum cap (for stability purposes) by adding a small resistor in series with a ceramic cap. For example 0.1 Ohm or even 1 Ohm. Just find the ESR of a recommended tantalum cap and use a resistor of that size. Do a stability test when you build the circuit and tweak the resistor value if needed.
As far as ceramic input bypass capacitors go, it is worth reading about the hazards of input voltage overshoot. See Linear Technology App note AN88. It is well worth your time. https://www.analog.com/media/en/technical-documentation/application-notes/an88f.pdf. The quick summary is that the cord from an external power source acts like an inductor. Your ceramic input capacitor in conjunction with this inductor forms a high-Q LC which can overshoot surprisingly high when power is first connected to your board.