Why do the Darlington or Sziklai configurations need complementary transistors?
They don't. In fact, it's usually not even desirable.
However, before we get too far, let's make it clear what we are talking about:
In this example, both constructs result in overall NPN polarity. I'll leave it as an exercise to the reader how to make overall PNP results, but want to make it clear there are really 4 total variations taking all polarities into account.
There is no point in a "complementary pair" for use in the Darlington configuration, since both transistors are of the same NPN/PNP type. If you wanted the same properties, you'd just use the same transistor.
In both cases, the two transistors are cascading. The first amplifies the input current, which is then used to drive second. The second drives the load. The second transistor (Q2 and Q4 in the examples above) is therefore intended to take the bulk of the current. Since the requirements of the two transistors are quite different, you usually don't want them to have the same specs.
It might be perfectly appropriate for the first transistor to be a small signal type, and the second to be a power transistor.
About the only time when you would use the same or complementary transistors is when they are handling signals instead of power, and pretty much any transistor could do it. In that case, it is often convenient to specify small signal transistors already used in other parts of the design, or your common "jellybean" transistor. For example, for low current signal applications, I'd use 4401/4403 for my NPN/PNP unless there was a reason not to. Doing this across enough products reduces stocking requirements and allows buying the parts in larger quantities.