Why would ALU Output be Asynchronous with Input Change?
Synchronous logic refers specifically to that with registers whose values are changed at (synchronous) to a clock edge. The registers act as a memory, so the output is not necessarily reflective of the current state of the input.

Asynchronous means there is no clock, and so the output, as stated, depends only on the inputs. There is a delay, which is due to the finite time it takes for the signal to propagate through the logic path, but that is not why it is referred to as asynchronous logic.