I think names for fields are largely defined by social forces behind it not its ideas. Terms like "statistics" are sociological artifacts, not mathematical ones. The difference between stats and machine learning is analogous to say, the difference between Canada and the United States. They're neighbours, separated at birth, share many of the same values but still each have their own personality, worldview, goals and culture.
Walk into a statistics department in a university, and you can feel the differences in culture. It seems to boil down to things about what one considers sacred. Confidence Intervals, Hypothesis Testing and Rigor are things statisticions hold near and dear to their hearts. Want to be a statistics phd? You'd better understand all the differences between convergence in probability and convergence almost surely. And things like Probability Approximate Correct Learning, VC Dimension, Random Forests, and even Deep Learning should sound like heresy to a true, card carrying, statistician.
Another odd cultural difference. Statisticians love citing old papers. You can call a paper about 20 years old "the classical work of" in ML. Stats has a much higher standard for this. The classics go way, way back to a time before I was born. Its gotta be an issue of pride.
But yea, I do feel for the most part, I feel the fields are converging, even if it seems like ML is consuming statistics, not the other way round. Stats majors are publishing in places like NIPS rather than the traditional journals, not the other way round. Just a feeling though. I have no stats to back up this claim.
But there's no point saying "A is a subset of B" unless you want to spice up your happy hour conversations in the pub.