The CMS muon detector system, muon reconstruction software, and high-level trigger underwent significant changes in 2013-2014 in preparation for running at higher LHC collision energy and instantaneous luminosity. The performance of the modified system is studied using proton-proton collision data at center-of-mass energy $\sqrt{s}=$ 13 TeV, collected at the LHC in 2015 and 2016. The measured performance parameters, including spatial resolution, efficiency, and timing, are found to meet all design specifications and are well reproduced by simulation. Despite the more challenging running conditions, the modified muon system is found to perform as well as, and in many aspects better than, previously. We dedicate this paper to the memory of Prof. Alberto Benvenuti, whose work was fundamental for the CMS muon detector.