Why use a tantalum instead of a ceramic capacitor?

The LDO built into the chip needs an ESR that is high enough to ensure stability under all conditions.

You can use a 1uF ceramic capacitor with a series resistor of 5 or 10 ohms in place of the tantalum capacitor. And keep the 100nF capacitor in parallel.

It's pretty much legacy information. For instance, the AD7192 on page 36 says the same thing and that dates back to 2009. The AD7124 is 2015 vintage and it appears to have just copied over some of the same words: -

If I could be bothered to go back further I'm sure I'd find that pretty much all similar devices that are older say the same thing.

You can use MLCC capacitors.

MLCCs have mostly better ESR and ESL compared to tantalum capacitors, but tantalum capacitors are a bit stronger against voltage and temperature fluctuations. For example, an MLCC's capacitance may change 40% at 50% of rated voltage.

Plus, some converters need a bit high ESR and a bit high ESL for stabilization (e.g. to prevent unnecessary oscillations).