French and English use “brand new/neuf” and “new/nouveau” in much the same manner, so don’t sweat this.  In both languages, there is a way to emphasise something is ‘new’ as well as ‘new to the owner’ when desired - ‘neuf’ or ‘brand new’ does this. So, there is little linguistic difference here. Just remember ‘neuf’ will always evoke ‘brand new’ in a French speaker - if you describe  “une voiture neuve” and they see a beat up car, they will be very surprised !

We would usually not say ‘bought a brand new hat/shirt’ etc in English - nor is it common in French. In either language if the context left doubt, further questions and comments would be used to clarify. It is not wrong to say it if you mean to, though.

We are even less likely to say ‘brand new’ (boy/girl)friend, partner, spouse, dog/pet or (anything living), and this is true in French as well. Less likely = almost never.

The important difference is the grammatical one - French routinely uses ‘neuf’ after the noun (standard placement for most nouns), and ‘nouveau’ before the noun.

Supplementary information :   Other adjectives change meaning when before or after the noun (look up moveable adjectives eg prochain/ancien etc etc), even though the word stays the same - at least here there is also the small word change to make it possibly easier (for English speakers)!