How to realize multitrack signal playback in different sound cards separately

For PCs:

It's not possible to do this with an ASIO driver - the system was designed by Steinberg so that it would only address one device - nothing to do with Audition at all, that's their limitation and any software using it is forced to abide by it. The reason is that it punches a hole clean through most of the OS, and consequently performs rather better. It works well with Audition, as Audition has an ASIO-based engine.

Because of the way most other device drivers work, I think that there's only going to be one way to do this, and that's to use ASIO4ALL, which will allow you to aggregate more than one device, but the whole setup will have higher latency. For playback only, that's unlikely to be a problem. ASIO4ALL is free, and relatively easy to set up. It also has good diagnostics, so you can tell what's happening and whether you have any problems you need to address.

For Macs

My understanding is that Macs aggregate audio devices anyway, so it's a matter of getting them connected up and the system configured correctly, I guess - I don't have one so I can't tell you absolutely what you have to do.