Yes, you can use the same code. It is much easier to do this than it is to get a second account. From the AdSense FAQ:
4. Can I show ads on more than one site? Yes, and you don't even have to let us know because we'll automatically detect new sites that contain the AdSense ad code. The HTML code we provide you with is valid for any page in any site on which you want to display AdWords ads. However, you may only display AdWords ads on sites that you own or on sites that you are legally authorized to act on behalf of for the purposes of Google AdSense. We also monitor all sites in AdSense to ensure that all pages displaying AdWords ads comply with our program policies.
You can find the FAQ here: https://www.google.com/adsense/faq It is well worth a good read-through, and it tells you what you can and can't do, along with answers many questions about the program.
Part of the benefit of AdSense is that it targets within a few minutes. So you could have a site on X and have ads all targeted just fine and then put the code on a site about Y and within a few minutes be serving Y ads. No muss no fuss. Don't even have to set up keywords!