Earlier quoted context omitted.
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.
The most common solution at the time was PulseAudio, which was so bad it usually was better to just use direct ALSA and live with the idiotic one-at-a-time limitation.
Thankfully Pipewire seems to actually work reliably so I guess that's at least one thing ticked off the Year of the Linux Desktop checklist.