It’s not that they embed probabilistic behavior per se. But more like they are chaotic systems, in that a slight change of input can drastically change the output. But ideally, good compiler design is idempotent — given the same input, the output should always be the same. If that were not generally true, programming would be much harder than it is.

No, it can also vary on the input. The -ffast-math flag in gcc is a good example.