More or less--they're more like shock collars, which can be deactivated if necessary. So technically a mage can use unauthorized magic while collared, it's just extremely painful. They also work as identification of, through color, what thread/element a mage works with and how dangerous/powerful they're considered, and, by tags, how long they've been a slave, whether they were enslaved in lieu of a prison sentence (before Sorell changed the law), what that crime was, etc.