It's often an artistic choice. These days, when digital recording is the standard in studio's, many have "tape emulators" that add the tape distortion to sound because that is what is wanted. In electronic music, bitcrushing drum tracks is often done as artistic choice. And colouring trough tubes, transformers, semiconductors (mostly FET's) and opamps are common in recording of pop and other modern music styles.
This is because a music production is mostly not recording what is played live, but a assembly of music, that is made to sound a certain way how the artist or the producer wants it. That has always been the case since multitracks were availeble. It's only for classic music, and some styles of jazz that they try to stay close to the original sound of the instruments in the room. The rest is heavy processed with eq, compression, reverb, delay ao tools to sound in a certain way. And mostly it's not recorded in one take, but in several takes and edited and mixed later.
Some tracks are even not fully recorded in the same studio. This song was largely recorded in the UK by Isobel Campbell (including some session musicians), while Mark Lanegan recorded his parts (mainly some of the guitars and his vocal parts) in the US. They only met physically after the album where it was on (Ballad of the Broken Seas) was done fully trough the internet connection they had...