Here is an analogy:
"If I take something from somewhere but tell everyone where I took it from, is it against the law?".
Taking something from somewhere is not against the law. But if the owner of the thing does not like what you did, that is where the problem starts. If you take a small chocolate from a supermarket and in return you offer to carry a big banner promoting the shop, they will be happy to let you take a chocolate everyday. But if you take an expensive laptop from a shop and in return offer to put a small sticker about their shop behind your car, that may not be enough for them to allow you to take away their expensive thing. In either case, police will not interfere with it until the shop complains.
The same rule applies here. AdSense does not care whether you took from somewhere or linked back to them. The problem starts when the site owner or the web community thinks you are "stealing content". Some small sites may be happy if you republish their content and give a backlink because they may be dying to get backlinks. But many other sites may not want others to take their content and republish elsewhere even if it accompanies a backlink.
The short answer is, if the other site allow you, you can republish content. HOWEVER, if your site contains no unique content and all that you have is reproduced content, Google will not see any value in the existence of your website and eventually it will be eliminated from AdSense and possibly from AdSense too.