minecraft-java-edition
On the minecraft wiki entry for monster spawners, there is this statement:
A monster spawner is only active when a player is within a distance of 16 blocks from the spawner (where distance is calculated the standard (euclidean) way)
Does this calculation take into account vertical distance as well as horizontal distance?
The conditions shown in your screenshot should be able to spawn pigs, assuming it's a pig spawner. I did a rough recreation of your setup in my world.
Even in the dark, pigs were spawning, and within seconds of placing the spawner, so it's not as if you aren't waiting long enough. (Yes, the pigs also spawned when I surrounded the spawner with Glowstone). Because of this, I have a feeling that the spawner in your image isn't actually a pig spawner; it's likely a zombie spawner, or some other hostile mob. This is especially likely if you found the spawner and it wasn't manually placed by anyone, since there wouldn't be any natural pig spawners around.
Like Kevin Reid said, you could test this by making the room dark and seeing if any hostile mobs spawn. If you have admin privileges (and don't feel like this is cheating in some sort of way), you could use /give 52 1 to give yourself an actual pig spawner, place it in the room, and see if that one spawns pigs, just to verify if this is an odd SMP glitch that is only happening for you.
Looking at the source code, near a portal means on the base of the obsidian portal frame. The pig zombies will also look in a completely random direction.
Best Answer
To test this, I built a tower over some spawners and stayed directly over them about 15-20 blocks higher than they are.
As you can see, the flames have disappeared (and the spawners quit spinning, but you can't tell that from a picture), indicating that they are no longer spawning, despite me being directly over them.
So yes, it does take into account vertical distance.