I came over a bug last month that enabled me to legitimately record a video on my iPhone's camera app without having to pause the music that was playing. But I shut my phone off and back on and it stopped working, I wish someone knew how to trigger this again or inform me if it was patched by apple in a recent update, but yea, that's basically it.
You can't currently record a video while playing music using the native app.
Good news is that you can use 3rd party apps to do it instead, here is an example.
I use the iPhone app Mideo: Record and Listen and it doesn't pause my music (Spotify, Apple Music, SoundCloud, etc) when I record video. Disclaimer: I helped develop this app.