Consider a barbarian that loves to charge at things. On top of his trusty axe (which is indeed a manufactured weapon) he also has a pair of claws by the virtue of Beast Totem or maybe from some another source. He then gets on a mount.
Spirited Charge says this:
Benefit: When mounted and using the charge action, you deal double
damage with a melee weapon (or triple damage with a lance).
Now, if the guy charges and attacks with his axe, he obviously deals double damage. What if for some reason he decides to attack with his claws? Is the damage doubled or not?