  The $s$-channel resonant production of the Higgs boson at a 125 GeV muon
collider enables a unique way to determine the Higgs properties. However, a
clear picture of the achievable Higgs precision has not yet been established.
We perform a phenomenological study of the Higgs measurements at such resonant
muon collider Higgs factory and present a systematic, detailed, and consistent
extraction of Higgs precision measurements. Many new aspects about the
lineshape scan, including the scaling with luminosity, optimal scan range,
minimal scan steps, correlations with exclusive measurement, effective
cross-section modeling, etc., are quantitatively studied in this work. All
major exclusive Higgs channels are simulated and analyzed with Standard Model
background, detection efficiencies, acceptance, angular distributions, and
cross-channel correlations. Global analyses of the Higgs couplings are
performed in the $\kappa$ framework and the effective-field-theory one. The
results suggest that the 125 GeV muon-collider Higgs factory provides
significant improvement to the Higgs coupling reach of the HL-LHC and provides
independent and distinct Higgs precision information concerning future $e^+e^-$
colliders. We report results for both $5~{\rm fb}^{-1}$ and $20~{\rm fb}^{-1}$
integrated luminosity. These results provide comprehensive and quantitative
physics understandings helpful in planning for the muon collider roadmap and
global high-energy physics programs.
