Is a SSR designed for AC power fundamentally different from one for DC power
Yes they are different. The SSR acronym is mostly used for the AC switch which uses triac or thyristor pair as the switching element. Triac can conduct in both directions, it starts conducting if the voltage is enough high and when a gate pulse is issued. It stops conducting when the current falls bellow a certain threshold. The gate pulse has to come each semi-cycle as the triac goes off at every zero current.
The DC solid state is usually made of MOSFET or IGBT transistor with appropriate gate driving circuit. The are more expensive, not used often since the power used is usually AC.

I know that PID control of AC power requires zero-crossing detection
and other complicated things. This doesn't seem like it ought to
prevent a SSR from switching DC power.

They are rather simple and cheap devices. An AC SSR would never turn off, if you apply DC voltage. There are also phase angle control SSR, they are not zero-cross and they don't conduct the whole cycle, rather an arbitrary conducting angle. Such devices are usually controlled by 0-10V (or 4-20mA) signal from PID controller.
Beside zero-cross and phase angle SSR, there is also a random phase AC SSR. This doesn't have a zero-cross detector, it starts conduct at any time. The  ZCD can be used for driving a resistive load, like heaters. Meanwhile with inductive load you should use a random phase SSR or phase angle ctrl, as the ZCD would skip periods if voltage/current are not in phase. If the used  SSR is not ZCD, then an additional EMI filter is needed.