What do we mean when we say it today?
We don't see any twins confusing their twin servants for each other in real life. (If you do, send them our way, because we know a Shakespeare role that would be perfect for them.) Even so, we hear this phrase tossed around all the time.
When someone says there's "no rhyme or reason" today, it means the same thing as it did when S. Dromio first said it—minus the twin stuff. It basically means that there's no rationale for whatever's going on because there's no pattern (rhyme) or logic (reason) behind it. You just can't figure it out no matter how hard you try.