It does. Why? Because of tape saturation. Back in the day, most recordings one multi track machines were overbiased at +4, then the rock crowd went as far as +10 to get a hotter track. This is were the "warmth" comes in, but what it really is is a sort of compression and the magnetic particles becomes over saturated and can't handle any more signal. Hence, that tape sound.
You don't get this with digital but there are plug ins that emulate tape saturation, not too well, in my experience. There is a faction of engineers who record to tape, then sent the post FX tracks to digital for final mixdown. I have tried it with an Ampex 2" 16 track (flat top) and it is definitely warmer. The 16 tracks onto 2" give a wider span per track to record onto, vs 24 tracks to 2".
As far as vinyl one of the things that affect the sound is the use of the RIAA curve. The needle can only track so much, so, when the master is cut, the highs and lows are pulled back, so the needle has less to deal with - less skipping and bounce. The highs and lows are restored (RIAA curve) in the phono in preamp stage.