We study the discovery and discriminating prospects of the Higgs portal dark matter (DM) models for scalar, fermion and vector DM at the ILC with $\sqrt{s}=500$ GeV and at proton-proton (pp) collider with $\sqrt{s}=100$ TeV. For the singlet scalar DM, the mediator is just the Standard Model (SM) Higgs boson, whereas for the fermion or vector DM there is an additional singlet scalar mediator that mixes with the SM Higgs boson, which produces significant observable differences. At the 500 GeV ILC, the DM pair is dominantly produced in association with a Z boson. The distributions of DM pair invariant mass can be reconstructed. It is used as the main handle for discriminating the DM spin. At the 100 TeV $p$-$p$ collider, we consider the $t\bar{t} +$ DM associated production in dileptonic final states. The two dimensional distributions of missing transverse energy and polar angle difference between two leptons are used for a discrimination of the spin nature of DM.