There are several questions to be asked here.
First, there is: "Can prestidigitation change the colour of the coins so they are golden?" The answer is yes. It's quite easy.
But there is another, more important question. "Are gold coins exactly the same shape as silver coins, just made of gold instead of silver" Ultimately, this is up to your DM, but I think that gold, silver and copper pieces look different, and not just due to coloration. They have other designs on them, and maybe even something like "1 gold piece" engraved. The reason for that is two-fold (at least, but that's the two reasons I can think of right now). 1. If you could just change the colour of the coins, you would have more forgeries than coins. Those with magic ability would use prestidigitation, and the others would get gold paint. So we give them different designs, with kings adoring gold pieces, princes on the silver ones, and coppers for barons. Or any other design. 2. Creatures that live in the sunless deeps and rely mostly or solely on darkvision use coins, too. I doubt that drow want to bring a candle to look at their coins, especially if they want to conduct illicit business, and that's one of the most favourite hobbies for drow.