On an analog phone line (two wires) the transmitted and received audio coexist. Obviously the transmitted audio is far, far louder than the received. The hybrid (the parts in the handset) subtracts most of the transmitted audio from what you hear in the earpiece, leaving only a bit of "sidetone". Too little of that, and the phone appears dead. Too much and you distract yourself with your own voice.
Many of them do. My Android 9 phone has it.
The feature only turns on when your calling an actual phone number though, voice calls like Discord don't trigger it.
It's not analog, but it's a feature built into the audio chip so there's no noticeable lag. AFAIK, it's not exposed to other software at all, because even the official Google Sound Amplifier app has noticeable lag since it does it in software.
My Samsung S10e (Android 12) doesn't do it—I made sure before making that comment. I didn't know that some Android phones do have sidetalk. In my experience talking on a landline, sidetalk makes talking much easier for me, more real somehow. Talking into my cellphone is like talking into the void, and I avoid it whenever possible.
I've never actually listened for it, but for sure audio calls (on a Moto G Play 2021 in my case) are less than satisfying. As a result I nearly always (except when privacy matters) use it in speakerphone mode.
https://en.wikipedia.org/wiki/Telephone_hybrid