If you dump oil into the sink, it ends up in the sewer system. As it cools, the fat particles stick together and solidify. When combined with things like wet wipes or tampons, it forms massive, rock-solid clogs that can even push sewage back up into your sink.
A better option is to place some paper towels in an empty container and pour the grease over them. The paper will absorb the fat, and once it cools and hardens, you can simply throw it away with your regular trash.