You can use the YouTube music app with a standard (free) YouTube account, at least on iOS. But the audio only keeps playing in the background if you pay for YouTube music. And I think you can’t download music to play offline.
For what it’s worth, YouTube Premium ($14/ month) includes YouTube music and removes ads for YouTube.
Having YouTube premium means I can use it on any platform. I can use YouTube music on my phone with CarPlay and listen while driving, use the built-in YouTube app that hotel TVs have when I travel for work, watch things on my work computer, on my phone, or using my PS5.
Its a but hacky, but if you disable the youtube app, install firefox and then its "video background player" addon, you can listen to youtube videos/music minimized without paying.
Seems to vary depending on whether I'm attempting to listen to music on my device ("attempting to" is the right word as it takes 15 seconds to load each individual song as it goes through a playlist) or listen to a YouTube video. It can go further into the background if it's music on my device, no further back than a big window drawn over the screen if it's a podcastable video, and must be in the foreground for a video IIRC.
The current Podcasts app does, but Youtube doesn't unless you pay for premium. After the migration, if that policy doesn't change, google's platform of the week will be useless for podcasts.