You could do something like this:
struct Product {
		var brand: String
		var option: [String]
		var item: [String]
}
let products = [
		Product(brand: "Academia", option: ["Blue", "Red"], item: ["academiaBandagesBlue", "academiaBandagesRed"]),
		Product(brand: "Academia", option: ["Purple", "Green"], item: ["academiaBandagesPurple", "academiaBandagesGreen"])
]
var tackInventory: [ String : Int] = [
	 "academiaBandagesBlue" : 1,
		"academiaBandagesGreen" : 0,
		"academiaBandagesPurple" : 0,
		"academiaBandagesRed" : 0
]
let filteredKeys = tackInventory.filter { $0.value > 0}.map {$0.key} // Select the keys to search for
print(filteredKeys)
for key in filteredKeys {
		let filteredProducts = products.filter { $0.item.contains(key)} // all the products that contain that specific item
		for product in filteredProducts {
				if let itemPosition = product.item.firstIndex(of: key) { // What is the position of key in item list (there should be only one ?
						let foundColor = product.option[itemPosition] // If color are in the same order than item, then we got it
						print("key", key, "color", foundColor)
				}
		}
}
and get:
["academiaBandagesBlue"]
key academiaBandagesBlue color Blue
But your coding of Product is not robust at all.
If you do not put color and item in same order, you get wrong result.
You'd better have a dictionary linking item with its color.
You could also define item as a couple of String:
struct Product {
		var brand: String
		var item: [(color: String, ref: String)]
}
let products = [
		Product(brand: "Academia", item: [("Blue", "academiaBandagesBlue"), ("Red", "academiaBandagesRed")]),
		Product(brand: "Academia", item: [("Purple", "academiaBandagesPurple"), ("Green", "academiaBandagesGreen")])
]
var tackInventory: [ String : Int] = [
	 "academiaBandagesBlue" : 1,
		"academiaBandagesGreen" : 0,
		"academiaBandagesPurple" : 0,
		"academiaBandagesRed" : 0
]
let filteredKeys = tackInventory.filter { $0.value > 0}.map {$0.key}				// Select the keys to search for
for key in filteredKeys {
		let filteredProducts = products.filter { product in
				for item in product.item { if item.ref == key { return true }}
				return false
		} //all the products that contain that specific item
		for product in filteredProducts {
				for item in product.item where item.ref == key {
						let foundColor = item.color
						print("key", key, "ref", item.ref, "color", foundColor)
				}
		}
}
and get
key academiaBandagesBlue ref academiaBandagesBlue color Blue
