As far as I know, metallic hydrogen and plasma seems share some common properties, eg:conduct electricity, nuclei share electrons, high temperatures...
So my question is, is metallic hydrogen in fact plasma state of hydrogen? If not, what is the difference between metallic hydrogen and plasma of hydrogen?
The primary difference is that the electrons in metallic hydrogen are nearly completely degenerate.
Degenerate electrons cannot be dissipatively scattered and lead to the "metallic" characteristics of extremely high electrical and thermal conductivity.
To first order, to produce metallic hydrogen you need to make sure that the electron kinetic energy at the Fermi energy of the system is much greater than $kT$. For a pure hydrogen gas and non-relativistic degeneracy, this leads to $$ \left(\frac{3}{8\pi}\right)^{2/3} \frac{h^2 n_e^{2/3}}{2m_e} \gg kT$$
This means that either very high electron densities, or low temperatures, are required.
"Metallic hydrogen" can exist as a fluid or a solid. In liquid metallic hydrogen, both the protons and electrons are "free". To produce a solid requires that the ratio of the coulomb energy of the protons is much larger than $kT$ so that they "freeze" into a crystalline lattice. This in turn is a requirement that needs even higher densities or even lower temperatures. Thus under these conditions the electrons are indeed completely degenerate,
In metallic hydrogen, the protons share an approximately fixed location relative to each other - energetically, a lattice is more favorable than an randomized state. Because of this, the substance is not a plasma - in a plasma, the positive and negative charges both flow freely.
Metallic hydrogen only exists at very low temperatures, and very high pressures. According to the phase diagram here, pressures above 2 Mbar, temperatures below 100 K (note the dotted line - this is a very fuzzy region of the phase diagram).
