Looking at the diameter of the actual projectile (which is what the size of the weapon refers to) the closest thing I can find is the
British 4.5" AA gun, which is consistent with the length of the casing and the height of the projectiles in the pictures at the link. AA guns needed high velocity becuase of the speed of aircraft and the height their rounds needed to attain, hence the long casing with lots of powder. I
believe this is an example of the British AA round.
As for Lex, just ignore him. Aside from the fact that 8" weapons would almost always be the main battery for heavy cruisers in WWII, but rarely, if ever, land artillery, you would not be "shelling your own forces" if you used an 8" gun for AA because A) most of the time the ship is going to have the round go into the sea if it does that and B) even if you do use it where land is around, the chances of it landing on your own troops is remote. Generally when troops get hit by friendly artillery fire it's because of very specific mistakes that accidentally target them. Random rounds are most likely just going to blow up dirt unless they land in a pretty urbanized area.
As for why someone would use an 8" round on aircraft - yes, it's expensive. A heavy cruiser is a lot more expensive, however, and you only have so many of them and they take a lot of time and resources to build. If you can elevate your main battery high enough to possibly hit an enemy plane, it may be worth it to take those off chances. American light cruisers often used their 6" main batteries against aircraft, occasionally with success, and
Yamato supposedly even tried using it's 18" main battery against aircraft at LEyte Gulf. The real question is, is the air threat severe enough that onboard ammunition depeltion is justified. The cost of an 8" shell in the middle of WWII is... not a major factor, especially since gun rounds are really pretty cheap.