  We study the discovery potential of the non-Standard Model (SM) heavy Higgs
bosons in the Two-Higgs-Doublet Models (2HDMs) at a multi-TeV muon collider and
explore the discrimination power among different types of 2HDMs. We find that
the pair production of the non-SM Higgs bosons via the universal gauge
interactions is the dominant mechanism once above the kinematic threshold.
Single Higgs boson production associated with a pair of heavy fermions could be
important in the parameter region with enhanced Yukawa couplings. For both
signal final states, $\mu^+\mu^-$ annihilation channels dominate over the
vector boson fusion (VBF) processes, except at high center of mass energies
where the VBF processes receive large logarithmic enhancement with the increase
of energies. Single Higgs boson $s$-channel production in
$\mu^+\mu^-$-annihilation via the radiative return can also be important for
the Type-L 2HDM in the very large $\tan\beta$ region, extending the kinematic
reach of the heavy Higgs boson mass to the collider energy. Considering both
the production and decay of non-SM Higgs bosons, signals can be identified over
the Standard Model backgrounds. With the pair production channels via
annihilation, 95\% C.L. exclusion reaches in the Higgs mass up to the
production mass threshold of $\sqrt{s}/2$ are possible when channels with
different final states are combined. Including single production modes can
extended the reach further. Different types of 2HDMs can be distinguishable for
moderate and large values of $\tan\beta$.
