Strange passage in Whiterun

It's in all probability a mistake.

Toggling collision using tlc in the in-game console, I checked the location you mentioned, and found the 'random passage':

Screenshot of the location in-game ^{View from underneath the city wall, facing the House of Clan Battle-Born in Whiterun, and showing the text 'To'. Mind that it's not connected to the water outlet.}

Going 'through' it, I was also shown a black loading screen (with only the mist or smoke), with the sounds continuing in the background. So we need to look for answers elsewhere.

In the Construction Kit we can check what entity causes this behaviour:

Screenshot of the mentioned place from within the Construction Set ^{To the right we again can see the House of Clan Battle-Born in Whiterun. The red cylinder shape left of the middle is the entity responsible for the 'Activator' showing up in-game. The green cylinder, and yellow arcade/door in the centre, are part of the entity.}

The entity is a Reference, with Reference Editor ID 000351EC.
It refers to a Door named "Passage", with the ID "AutoLoadDoorHiddenMinUse01" (an entity used 5 times in the vanilla base game ^*).
It has no Teleport data, meaning it leads nowhere - which probably explains why it only shows the word 'To' in-game. Nor does it have any other references or scripts to give us a clue to its purpose.
Its checked flags are 'Hidden', 'Automatic Door', and 'Minimal Use':

Screenshot of Construction Kit showing the dialogue boxes of the entity

According to creationkit.com, the 'Hidden' flag should - not surprisingly - hide the Door in-game.
'Automatic Door' means it acts like a portal, and requires no interaction.
'Minimal Use' makes sure NPC's will try to navigate around it.

I couldn't find any mention of the Reference ID through Google, and a quick skimming of the Unofficial Skyrim Patch changelog doesn't give any results either.

My guess it has no purpose (anymore), and can very well be a left-over from development.

It could also still have something to do with the Civil War changes that a playthrough can bring about (linked to Whiterun being under attack), but that's nothing more than a hunch.

^{* For those wanting to gaze even deeper into this abyss: below this is the 'Use info' popup of AutoLoadDoorHiddenMinUse01. The other references have active 'Teleports'.}