With regulators (linear or switcher), the output capacitor can interact with the regulator’s feedback behavior and cause it to be unstable. This is mainly an issue when ultra-low ESR ceramics are used in place of electrolytic or tantalum. Otherwise it’s ok to use an aluminum electrolytic or polymer type in place of tantalum, they're similar enough.
Read further into the datasheet - it doesn’t require a tantalum. An aluminum electrolytic is adequate. They recommend tantalum only if very large load transients are present.
Which leads me to, why is tantalum in their reference design? Habit, mostly. They probably used tantalum to show the very best transient response for their part. Likewise with your inductor example: a reference design used it to best show off the part. Either way, for your design you can make different trade offs for cost, space, efficiency, etc.
That said, tantalum caps have some advantages, especially in density, stability and low leakage. They also have downsides:
- Tantalum is a conflict mineral and should be avoided.
- Tantalum caps can catch fire when overstressed (overvoltage, reverse voltage, temperature.)
- Tantalum caps are kind of expensive compared to other options.
For these reasons I haven’t used a tantalum in a design since the 1980s.
What about those 'other options'? Aluminum polymer electrolytic caps offer similar transient-handing benefits to tantalum without its drawbacks. With care, some designs can even migrate to MLCC.
More cap selection stuff here: https://resources.altium.com/p/which-type-capacitor-should-you-use
For their part, IC manufacturers have also responded and have reduced the need for tantalum caps. Some voltage regulators are stable with MLCC (ceramic) output caps, or can be made stable if designed carefully (such as adding a small series resistance to the cap to model the ESR needed for stability.)
Meanwhile, MLCCs have become more dense, offering superior high-frequency bypass performance to tantalum which allows downsizing bulk caps from what would be required if they were tantalum or electrolytic. This is especially the case for high-frequency (1MHz or so) switchers which are often designed to work with ceramic caps.
More about migrating to MLCC here: https://product.tdk.com/en/techlibrary/solutionguide/mlcc_replace-guide.html