Do cavern edge tiles count as their own tile in the Legend of Drizzt board game?
Yes, they are their own tile.

The rule book defines Tiles and Squares in page 6:

A tile is the basic building block of the dungeon, drawn from the Cavern Tile stack.
A square is one of the spaces on a Cavern tile.

Unfortunately, this definition technically rules out the Cavern Edge Tiles, because the are not drawn from the stack. Still, on page 6, it still refers to them as tiles, just 1/4th the size.

Cavern Edge Tiles - Each of these tiles is about one-quarter the size of a regular Cavern tile. The most common use for Cavern Edge tiles is to create a cavern complex at the start of an adventure. See “Creating a Pre-Built Cavern Complex” on page 14.

Since the rules treat the Start Tile as two separate tiles that are already connected, I see no reason not to treat the Cavern Edge Tiles as separate tiles connected to normal Cavern tiles.