The Bahamian dollar is exchanging 1 to 1 to the US Dollar. On top of this a lot of Cruise Ships originating in the US make a stop at Nassau.
Does that mean that I can use US Dollars in Bahamas just as I do Bahamian ones?
Yes you can. US Dollars are accepted by most every business in the Bahamas and some places, such as casinos, only accept US Dollars.
Wikipedia says that businesses in the Bahamas accept both Bahamian and US dollars. However, this is because they choose to do so and it's not an inevitable consequence of the exchange rate. Indeed, it's common for businesses in many parts of the world to accept multiple currencies even if there's no formal relationship between them: they do that because it is more convenient for them and their customers.