Pure ALSA would behave like that because the currently playing process would take exclusive control of the hardware.
Upsite: Highest quality playback.
Downside: Only one process could play audio at a time.
ALSA had dmix for some time already, but the default configuration of your distro may not have enabled it.
Most distributions shipped ALSA preconfigured with dmix, which means multiple applications could play sound at the same time just fine.
Which is why the whole "we must use pulseaudio even if it's terrible and has awful standards that blast volume or multiple streams won't work!" was so weird… everybody who tried knew that just removing pulseaudio the multiple streams kept working :)
So only those who never applied the scientific method kept insisting that without PA it was not possible to do that.
Only if you had no hardware nor software mixing configured, which probably should be considered a misconfiguration of your system.