If you look carefully at that game its actually entirely flat. Walls etc are drawn as floor tiles. Falling off the wall is then a case of: "if player moving from above: allow through and push to next tile, else, don't allow", end result, you enter a "wall" tile and the character goes to fall animation and gets moved to the tile below it. Same would work for falling off the building at the beginning. Think the pokemon GBA games, clearly flat and always had that gym with the floor tiles that shot you off in a direction and those dirt ledges you could walk down but not up.

Clever art makes it look like there is height to it but in reality its not.

But what would happen if I wanted objects that the player could walk over but also go under. for example a bridge. because one of the cities in my game is going to be pretty vertical.

You would have to have that extra Z-layer (assuming X and Y are used for the actual tile drawings). So, you could have tiles at simultaneous X,Y location, but different Z-levels. What I would do is allow a player to traverse across a Z difference of 1 (going up or down stairs), but, that's it. If you have multiple tiles at the same X,Y, but varying Z, the player would only be able to go go to the tile with the same Z, or within 1.

For example, if a player going up, starts at 1,1,1 and ground tiles are at 1, 2, 1 & 1, 3, 1 and you have bridge tiles at 0, 2, 5 & 1, 2, 5 & 2, 2, 5, the player would be able to move "under" the bridge from 1, 1, 1 to 1, 3, 1 (walking under the bridge at 1, 2, 1).

If a player is at 0, 2, 5 moving right, he would walk across the bridge at 1, 2, 5and end up at 2, 2, 5.

That's how I'd do it atleast, using walkable tiles, wall tiles, and air tiles.