I have reference recordings that sound better in vinyl and others that sound better in digital. Neither sounds more real to me (or the people I work with from my short survey here) just because it is one or the other, it is material dependent. Could be due to the different mastering, recording chain, or any other dozen issues. I can provide examples if you'd like. As such, I pose again the idea that this distinction (analog-digital) doesn't really track well with reality. There is nothing more continuous or natural about analog than digital, it is still stored media using a discrete resolution and an engineered encoding scheme.
As to why analog sounds so good, I have three ideas: a) low order harmonics b) noise and c) fetichism.
a) Low order harmonics are what you get from mechanical systems, and everything in analog is electro-mechanical. These mask away obnoxious higher order modes very effectively to our earing apparatus and are, again, compatible with what you find in a real live music scenario from things that the mics don't pick up, either because they get buried (remember the best mic in the world is about 14 bit resolving...) or because the recording eng deliberately filtered them away by using highly directional mics and other techniques. Effectively analog could be filling in the blanks, with distortion that we find engaging. This is by happy coincidence more than anything.
b) Noise is pleasant. We get nervous in an anechoic chamber. We evolved to take cues from background noise, and seem to be effectively very good at it.
To the point where adding given types of noise to recordings makes things more readable for us. If you really think about it, it's perfectly inline with tweaks that audiophiles do all the time. We're just used to confuse certain noise with silence. The start of a crescendo is much more fascinating to me on vinyl that usually on digital, and I'm willing to bet it is because of the noise that my brain doesn't interpret as such, unless I focus on it.
c) Then we come to the last. I'm ok with people pushing back, but I have very little doubt about it. Need to read up on psy studies a bit more to formalize this better. We all know our experiences are brutally filtered and colored by our state of mind. The act of preparing and putting on a record or tape is a ritual just like any other, and it has consequences. It is not for free, it is not convenient, it involves dedication and sacrifice, so it playing with our expectations is just, well, expected. Some people align immediately with the act of just listening. They are not searching for the next track on a stupid screen, they just sit back and lift off. The impact of this might be bigger than we expect of care to admit, at least it is for me the times I tried to analyze it.
I don't think so. The analog process seems to discard a lot of information (a stylus simply can't move fast enough for certain things, it does have inertia; a tape doesn't really record 20kHz cleanly, it is frequency limited), but what it does seems to be so aligned with our internal mechanisms, that we are can still have valid discussions like this one well into the digital age. Encoding something to bits is very efficient and errors are well controlled. We can discuss why a lot of ADCs and DACs simply suck, but if you reverse a) b) and c) you can probably derive a good fist order approximation to my opinion pretty quickly.