Nah, FSA is not a direct sequel, because the backstory for FSA says 2 occasions that Vaati terrorised Hyrule, but if Vaati remembered Link, we would have seen something to indicate this.
On topic, not they are not the same mirror
Twilight Mirror is merely a lock to the Twilight Realm with the big rock being the door.
The Magic Mirror can teleport the beholder into the Dark World, or the Light World, I can't remember.
The Twilight Realm is a very different place compared to the Dark World as it has a proper palace and a civilisation.