As Meek Mill once said: "There's levels to this shit."
A good portion of songs/chants are coopted from other sources. Whether it's taking a famous song and applying new lyrics (e.g. "Que Sera, Sera") or adopting another person's/club's song for your own (e.g. Tottenham adopting Sheffield United's "Everywhere We Go*"), there's a rich tradition of taking things you've heard somewhere else and making it your own.
But that becomes difficult when you're dealing with a song that's SO notable and SO associated with a particular team. For instance, even though it's a popular broadway song that exists and is enjoyed in all sorts of other contexts, could you imagine anyone trying to sing "You'll Never Walk Alone" outside of a Liverpool match?
I'd say the same thing goes for the "I Believe" chant. We can't adopt that, because it so clearly belongs to the USMNT, even if USMNT supporters didn't create it themselves.
*I've heard this coming out of our supporters section over the past couple months and it upsets me greatly.