In Hungarian, both are 'paprika'. But because the pepper itself is common outside of Hungary, most languages already have their own word for the pepper, and so the loan word 'paprika' is only used for the less common spice.

In Hungary, it's relatively uncommon to need to distinguish between the ground form and the vegetable form, as it seems to be some kind of folk knowledge imparted by generation DNA or something, but if you ever do need to, then the expression fűszerpaprika exists (meaning 'spice pepper').

We use the same word for bell peppers and black pepper, so it's not that big of a jump, just takes some getting used to.

No, US paprika is ground dried sweet or hot peppers. Often imported from Hungary.

It's not unique to english. In french, paprika is also used for the powder. While poivron is for bell pepper. And piment for the spicy variety. And piment doux for the sweet version.

Except it's not made out of the same variety of peppers as what we call bell peppers, according to Wikipedia.