How can I add a playlist to content type?
The aptly named AudioField module has the functionality you're describing


  Audio Field module adds new field that allows you to upload and play audio files in HTML5 audio player or one of selected Flash players. Currently it supports 6 different Flash players


It definitely works, I've used it, and it has support for both of the formats you mentioned. Follow the installation instructions and you shouldn't have any problems.

Just to nitpick, and perhaps clear up some confusion: Drupal cannot play audio files, no - your browser can, when it is given the right HTML (or when flash is used). But Drupal just generates the HTML, it doesn't have anything to do with the audio files themselves (beyond facilitating upload/storage) or how the browser interprets them.