It is perhaps unfortunate that they're called the Previous and Next buttons given it seems to lead to such confusion. “Back” and “forward” may have been better names. But in any case, the answer to your question is that ease of use and practicality trumps logic for this UX. You want a UI that makes the most common tasks easiest to do.
Among the most common tasks for users are:
To replay a song they’re listening too (e.g., because they got interrupted and didn’t hit pause, or maybe they just really like that song).
To skip the song they’re listening too (usually because they just don’t feel like hearing that song right now, not because they necessarily want to get to the next one, which may be unknown if they’re in shuffle mode).
Thus, it makes sense to have a couple buttons that support these tasks with a single press. It’s much less common for a user to want to skip back to a previous song or skip ahead two songs. It's good to support these tasks in some way, but also okay that they take additional effort (two presses).
Actually, “previous” and “next” are logical and consistent names, if you think of them as moving the user to the previous and next break between songs. However, I doubt users would naturally think in those terms.
BTW, I wouldn't say ease-of-use/task-support always trumps logic or consistency. It depends on how illogical or inconsistent the UI is. In this case it's a pretty mild inconsistency and readily mitigated. If any users expected Previous to go to the previous song, and found instead it went to the beginning of the current song, it would be easy and obvious enough to correct and recover by hitting Previous again. But it would be better still to rename the controls.