Why do linear voltage regulators have minimum output voltage > 0 V
Two reasons. 

1.25V is a convenient voltage to make a voltage reference- it's called a bandgap reference and has (relatively) low temperature coefficient at room temperature. You can make other kinds of references and you can make different voltages from a 1.25V reference with an amplifier or attenuator, but 1.25V is pretty good. You need a voltage (or a current reference, which is derived from a voltage reference usually) inside or you cannot regulate to a known voltage. 

Secondly, 1.25V is a voltage both low enough that few (until very recently) actually need a supply that low (really, nobody cares about lab supplies), and high enough that the offset voltage of the internal op-amp does not affect the accuracy much. It also allows an internal circuit that does not have to work down to 0V.

Making a simple adjustable voltage regulator that works down to 0V would not be particularly difficult in any way but it would add cost and pins, and that's a non-starter for a jellybean part.