From personal experience:
When people get upset, particularly in "matters of the heart", they feel it in their "chest", referred to as "breast" in times past. That is, they feel like they can't breathe or their heart is racing. This is base emotional pain, i.e., "savage".
And what do people do to "soothe the savage breast"? They listen to -- or write -- music. Ask any heartbroken teenager or song writer.
Most people have experienced this, far more than have ever tried (always unsuccessfully) to soothe a "savage beast" with music, so I don't see why the latter reference is more common, except that most people don't want to think about what sayings actually mean.
For example, most people think "possession is nine tenths of the law" means if you possess something, it is legally almost yours. What it really means is that most laws are about ownership, as opposed to criminal acts.