AD and AP aren't meant to be Fundamentally the same. If they were there would be no reason to have AD and AP or Armor and Magic Resist, you could simply call it Damage and they all work the same/get countered by the same things. Here are their differences.
AD- In total there are 25 full build items that give AD, if you build the items with the highest AD you can get 390 AD from just items. Ad in 5 Crit Only items so there are 30 AD only items in the game.
AP- Most AP Mages and Assassins can get to at least 700 AP there are 20 Full Build items that give AP.
The issue comes from what the AD items do, there are 2 Ad items that grant MR, 3 that grant a shield(Steraks, phantom dancer and hexdrinker). There are 3 items that shred Armor, it's not uncommon to get a BC+Last Whisper item so that you negate 70% of the enemy tanks armor. Then there's lethality so that you can shred up to 103 Armor. Ability Power has Magic Penetration, 40% from Void Staff and Flat Magic pen which caps at 39.
There's also Scaling differences, most AD champions do not scale with Total AD they scale with Bonus AD, but in many cases Bonus AD scales harder than Total AP. Here's a few examples.
Talon's Q scales with 110% of his Bonus AD, meaning if he has 200 Bonus AD he actually has 220 added damage to his Q. Scaling like this isn't uncommon by any means, most AD Assassins have Scalings like this on at least one ability. Talon can achieve this with 3 items, Duskblade, Youmu's and Ravenous Hydra or 8300 Gold.
Lux has a 60% AP ratio on her E, her most used spell. This means that in order to have the same extra damage on her E she would need to buy 338 AP. Lux can achieve this with 3 items, Morellonomicon, Luden's echo and Deathcap or 9800 gold. Technically she could achieve it with an extra item but building 4 items would cost a lot more than building 3. Hell even if she build Zhonya's, Banshee's, Morello's and Ludens she still wouldn't have 338 AP without runes.
This is what people don't seem to understand, the abilities of champions are Catered towards what type of Damage they build, it's not cut and dry and scaling is different for each one. To add AD has far more options Shields, Heals, Penetration, Defenses, Anti-Burst AD has it all while also having the highest scaling abilities on average and the fact that Crit is exclusive to AD champions and AD Damage. Mages and AP champions get Dodgeable spells, 2 niche defensive items, weaker overall defenses for Carry Champions since they can't build an inherent shield and less build path options in general.
Taking all of this into account, AD champions scale harder, have more ways to counter their defensive items and are overall far more adaptable, versatile and reliable than AP champions.