Does it mean that "I don't like what the President is doing just the same as you don't like what the President is doing"?
The negation of don't applies to the whole thing. Thus it means that the speaker likes "what the President is doing" to an equal or lesser extent compared to the person to whom they are speaking. The speaker does not like it more, so therefore likes it less, or likes it just the same. It is usually said to show they disapprove, usually to clarify when it sounds like they are defending the person (the President, in this case).