It's a huge leap (in mechanical assumptions) to go from, "the Background gives you one," to, "therefore it's reasonable to assume it can give you X many more."
Actually, you're ignoring the wording of the ability.
Anyone can use a Disguise Kit to create disguises and false identities and so on and so fourth.
Only the Assassin can do so unfailingly. And you'll notice the Assassin isn't using an Ability(Tool) check to create this identity; they simply succeed at creating one from using the ability.