I'm pretty sure USB-C adapters have tiny little DAC ICs embedded inside them. They show up to the phone as a USB sound card.
A lot of USB-C cables have little ICs built into them. It's required as part of the PD spec. A port isn't supposed to provide a laptop with 60W or whatever unless the cable itself communicates that it's rated for it.
There is both. Some phones, e.g. OnePlus, have analog audio output on USB-C and can use a passive adapter. If the phone lacks analog output, e.g. Samsung, it needs an active adapter with a USB controller and DAC like you mentioned.
Passing an analog signal fractions of a millimeter from a bunch of digital signals is a recipe for very crappy sound. I have yet to find an analog USB-C audio adapter that didn’t have terrible quality and usually some hissing or other noise too.
I have some balanced armature earbuds, and they are sensitive enough to pick up noise from poor digital grounding even in the 3.5mm jacks in a lot of phones. Additionally, they are often too loud even on the lowest digital volume setting. 8 years ago I actually made an inline attenuator, which solved the problem on my phone at the time.
My current phone is a pixel 6. I got a USB-C adapter with a DAC built into it, and it actually sounds pretty good. The volume levels are reasonable, and there's no noticable noise other than perhaps an unnoticeable hiss.
Years ago, I had a desktop case front IO panel with a really bad ground loop. I ended up cutting the molded over plastic open and cutting the PCB traces between the audio ports and the USB ports, and the problem went away.
That's how the USB-C to 3.5mm adapters work for if you need to use headpones which don't have a USB-C cable.