Background Sounds do not work on the HomePod mini.

Background Sounds are great for white noise, but they won’t play over a HomePod mini. If you want to use Apple’s built-in white noise to help you sleep or stay asleep, you would need to use another Bluetooth speaker or hardwired speaker.

I use the Bose SoundDock XT which is connected to my iPhone via an aux cable and USB-C to headphone jack adapter, and it simultaneously plays both YouTube and Background Sounds.

Same thing happens with my Anker Bluetooth speaker, where all sound is routed to the speaker.

If I turn off the Bluetooth speaker and select the HomePod mini as the source, the YouTube audio transfers over but the Background Sound stays on the iPhone. What’s even wackier is if I have my Anker Bluetooth speaker connected, I can play the YouTube video on the HomePod mini, and the Background Noise on the Anker speaker.

I know the HomePod mini uses both Wi-Fi and Bluetooth so it should be able to play both audio sources, but it seems Apple has prevented Background Sounds from playing on the HomePod mini, and I presume the regular HomePod as well.