It’s an old 19th century joke that Grant is noted to have told about himself. It’s a self-depreciating joke that implies that the speaker has very little musical ability or knowledge, being restricted to being able to recognize, reproduce, or follow along with [song name], and regarding all the rest as interchangeable or unreproducible. In this context, it’s meant to depreciate Grant’s own musical tastes as simple, undiscerning, or nonexistent (Yankee Doodle, of course, being a patriotic song and a very commonly heard one, and thus exempted from disinterest or unfamiliarity).

Sherman backs this claim of Grant’s lack of musicality - he said he himself knew only two tunes (Yankee Doodle and Marching Through Georgia), but that was “two more [tunes] than Grant”