The "at the same time" is the source of your problem.
Disabling Exclusive Mode might mitigate the issue.
https://wiki.jriver.com/index.php/Exclusive_Accesshttps://help.ableton.com/hc/en-us/articles/209770485-Disabling-exclusive-mode-for-ASIO-interfacesHowever, from your description, it sounds like you are running up against a fundamental limitation of audio hardware.
The sound degradation you report is probably due to resampling, or the windows mixer. Resampling occurs with DirectSound (which your browser uses) because audio hardware is only physically capable of playing one sampling rate at a time. Changing the sampling rate is really changing the clock rate on the audio hardware, and the clock cannot keep time at two different rates simultaneously. ASIO and WASAPI bypass the Windows mixer.
When JRiver is accessing the audio hardware to give you that good sound, the hardware cannot simultaneously play the browser audio which is at a different sampling rate. Using DirectSound resamples all audio to the same rate, which allows simultaneous playback of different audio streams, at the expense of quality.
If you turn off exclusive mode, MC will not be able to changing the sampling rate on the audio hardware, and you will probably think it sounds worse.
You could also try switching to WASAPI in MC. It should give you equivalent quality as ASIO (it depends on how good your ASIO drivers are). But the limitations with regard to exclusive mode will remain.
So, computer audio has limits. You'll have to chose which limitation you live with.
I hope this helps...