Mostly excluding tanks from this, because it sort of makes sense for both armor and magic resist to scale on them.
So yea. Was there a design reason behind it, or was it more just one of those oddities that came over from DotA?
EDIT: Specifically referencing the innate scaling with level, not scaling in general.
I think it's just because base AD scales with level, so it should be natural for the armor to do the same. (AP by contrast is built, so I guess that sorts makes sense?)
How do you define scale? If you already exclude tanks, than I'd say it anti-scales. The more defenses you buy, the less efficient it becomes. Your first 100 armor are much more fficient than when you already have 300.
Damage increases as the game goes on..
The idea is that you should be able to dive turrets at later stages of the game, so turrets deal less damage to you as the game goes on (this is also why turrets do not deal true damage).
Turrets don't deal magic damage so that's why MR didn't scale for the longest time (on ranged)
It's that Champions gain base AD per level, minions and monsters deal physical damage and also gain AD as the game goes on. Towers also deal physical damage and they don't want turrets to offer protection for the entirety of the game.
Finally, because there are more sources of physical damage and magical damage. What i mean by this is all champions deal guaranteed physical damage by virtue of basic attacks in addition to the AD champions with AD scaling on abilities. Without Armor scaling, AD would literally be king in terms of damage dealt as the game progresses.
There's several reasons, many of which have been answered already. But one reason I don't see mentioned is Level Advantages.
If I have a higher level than you (whether it be from killing you, zoning you from xp, farming more efficiently, etc), I ought to be able to win a duel with you (ceterus paribus). If stats no longer increased with level (including armor), Level Advantages would be less meaningful.
Trivia
One interesting oddity that was inherited from Dota is Attack Speed Growth. You'd expect the formula to scale linearly. But the formula is actually polynomial. However, the formula's parameters are tuned in such a way that the formula is extremely close to linear, to the point where the polynomial scaling is pointless. Riot could simplify their engine by linearizing the formula with negligible effect on gameplay. That is,
as = (as_per_level) (7 lvl^2 + 267 lvl) / (400) + items + runes + buffs
is practically identical to
as = (as_per_level) + 1.79 + items + runes + buffs
