When that Wikipedia article mentions an asynchronous interrupt, they are using the classical clocked vs. non-clocked definition of (a)synchronous, which applies to a digital circuit.
A digital circuit is said to be synchronous when every part of the logic is connected to a common clock (like in your CPU). At the rise or fall of every clock cycle, the state of the circuit is updated. An asynchronous digital circuit, on the other hand, is not clocked, but rather the next state is dependent on the current one (and will switch as soon as it can). Reading logic from other circuits that don't share the same common clock can also be defined as asynchronous, but with respect to the other circuit.
If an asynchronous interrupt is triggered, it means that the processor will (most likely at the next clock cycle) save its current executing environment, and service the interrupt request. This is an example of a hardware interrupt (one that is triggered by an external connection to the processor). All software interrupts, as mentioned in the article, are technically synchronous, since they are initiated by the CPU itself - which is a synchronous circuit.
Since no external devices share the same common clock as the CPU, all external interrupts can be said to be asynchronous. Even though the device that triggered the interrupt may be a synchronous circuit, from the CPU's point-of-view, those interrupts are triggered asynchronously (since it is not sharing a common clock signal with the device).
The actual external interrupt signal itself is asynchronous, but all CPU interrupt handlers are synchronous, they will only detect an interrupt on the next clock tick, since that's the point of a synchronous system (to only allow the system's state to change together). If you're curious as to how a processor handles interrupts, see this great resource from Intel (specifically, Volume 3, Part 1).