This answer is now obsolete. FlexASIO seems to be the current solution.
I think I know where is the issue. My Windows-fu is rusty, but here is what I found.
Solution, in short
Get ASIO multi-client from here or here (they are different, you might want to try both). After installation you should be able to use your interface with more than one program.
What is happening?
Seems that ASIO in general (including ASIO4ALL) can't handle more than one program using one same output.
Support for using ASIO with multiple programs at once. Previously, if you were using an ASIO driver
(so that you have low latency), you could only use ASIO with one program at a time. Now you can use
multiple programs at once with ASIO - for example RealBand and Band-in-a-Box at the same time.
Steinberg solved this by creating an ASIO multi-client server that you need to download and install. Seems that this is exactly what you need, and seems to work with all ASIO capable interfaces.
More info: http://www.pgmusic.com/forums/ubbthreads.php?ubb=showflat&Number=237170
There are other ASIO multi-client drivers you can try, like this one: http://vidance.com/asiomulti/asiomulti.html
From the above, we discover that:
ASIO (Audio Streaming Input Output) is a technology of Steinberg.
It allows for low latencies and pretty much every state-of-the-art
audio device is nowadays delivered with an ASIO driver
(on Windows, at least). However, the basic idea behind ASIO is
that professional audio applications entirely take ownership of
the ASIO device.
Only very few ASIO drivers support true multiple application
access. This effectively means that you can not use 2 or more
applications using the same ASIO device at the same time.
In other words, you'll need Steinberg's or Vidance's ASIO multi-client driver to route more than one program to the same audio interface.
Another alternative is to use JACK
I found some people were using JACK to solve this issue. It's an internal routing system for audio and MIDI. If it's anything like Soundflower, I believe they might be routing everything to JACK, and then routing JACK to the audio interface, using JACK like a mixer.
Edit: I've recently tried to use the Steinberg ASIO multi-client in Windows 10, and couldn't get it to work. The driver is really old and not officially supported, so it might be obsolete now (it probably works in other versions of Windows, but I can't test that atm).
What I found is that many audio interface manufacturers have a multi client built in in their ASIO drivers, so look for those specific interfaces if you want an easy solution.
Still no idea on how to solve this for ASIO4ALL though, other than the multi clients mentioned that might or might not work in different versions of Windows.