Is there a way to keep the folder structure while using the Apple Music app on iOS?

I assume you're talking about the Music player on iPhone - not the Apple Music Subscription service (which I don't use so don't know anything about).

Short answer - no you can't keep the folder structure, but.....

Go on your iPhone, open the Music app, navigate to Library and there are the "structures" you get": Playlists, Artists, Albums, Songs. You could, however, create playlists which mimic the categories you reference in your posts or the song orders in the folder structure. I use the Music app on Mac to create playlists then synch them with my iPhone which I connect to Carplay via USB. The only thing I use a different app for is shuffling because iPhone's Music app doesn't have album shuffle, just song shuffle.

If you want to go down this route then you'll need to import your music either into the Music app of a Mac or iTunes on a Windows PC, set up the playlists, choose what you want to synch and then then synch it with your iPhone.

There might be apps that do what you want but I've never looked for them.