Yes, and, no. Rum does originate from the Caribbean, but its roots are in the Bahamas. Therefore, to make a traditional Jamaican Mule, you ideally want to find Jamaican rum. While not all Jamaican rums are spiced, they are distilled in a specific way and therefore have a unique strong aroma and taste (you may have heard of the Jamaican 'funk').
Spiced rum is a popular choice for this cocktail as it packs a ton of flavour which, when combined with the ginger beer, is fiery and spicy making it a unique and delicious tipple that literally dances on the tongue.