  We analyze the physical properties of the gaseous intracluster medium (ICM)
at the center of massive galaxy clusters with TNG-Cluster, a new cosmological
magnetohydrodynamical simulation. Our sample contains 352 simulated clusters
spanning a halo mass range of $10^{14} < {\rm M}_{\rm 500c} / M_\odot < 2
\times 10^{15}$ at $z=0$. We focus on the proposed classification of clusters
into cool-core (CC) and non-cool-core (NCC) populations, the $z=0$ distribution
of cluster central ICM properties, and the redshift evolution of the CC cluster
population. We analyze resolved structure and radial profiles of entropy,
temperature, electron number density, and pressure. To distinguish between CC
and NCC clusters, we consider several criteria: central cooling time, central
entropy, central density, X-ray concentration parameter, and density profile
slope. According to TNG-Cluster and with no a-priori cluster selection, the
distributions of these properties are unimodal, whereby CCs and NCCs represent
the two extremes. Across the entire TNG-Cluster sample at $z=0$ and based on
central cooling time, the strong CC fraction is $f_{\rm SCC} = 24\%$, compared
to $f_{\rm WCC} = 60\% $ and $f_{\rm NCC} = 16\%$ for weak and non-cool-cores,
respectively. However, the fraction of CCs depends strongly on both halo mass
and redshift, although the magnitude and even direction of the trends vary with
definition. The abundant statistics of simulated high-mass clusters in
TNG-Cluster enables us to match observational samples and make a comparison
with data. The CC fractions from $z=0$ to $z=2$ are in broad agreement with
observations, as are radial profiles of thermodynamical quantities, globally as
well as split for CC versus NCC halos. TNG-Cluster can therefore be used as a
laboratory to study the evolution and transformations of cluster cores due to
mergers, AGN feedback, and other physical processes.
