  The behavior of H-like ions embedded in astrophysical plasmas in the form of
\emph{dense, strongly and weakly coupled} plasmas are investigated. In these,
the increase and decrease in temperature is impacted with a change in
confinement radius $(r_{c})$. Two independent and generalized scaling ideas
have been applied to modulate the effect of plasma screening constant
($\lambda$) and charge of ion ($Z$) on such systems. Several new relations are
derived to interconnect the original Hamiltonian and two scaled Hamiltonians.
In exponential cosine screened Coulomb potential (ECSCP) (dense) and weakly
coupled plasma (WCP) these scaling relations have provided a linear equation
connecting the critical screening constant $(\lambda^{(c)})$ and $Z$. Their
ratio offers a state-dependent constant, beyond which, a particular state
vanishes. Shannon entropy has been employed to understand the plasma effect on
the ion. With increase in $\lambda$, the accumulation of opposite charge
surrounding the ion increases leading to a reduction in number of bound states.
However, with rise in ionic charge $Z$, this effect can be delayed. The
competing effect of plasma charge density ($n_e$) and temperature in WCP and
ECSCP is investigated. A recently proposed simple virial-like theorem has been
established for these systems. Multipole ($k=1-4$) oscillator strength (OS) and
polarizabilities for these are studied considering $1s, 2s$ states. As a bonus,
analytical closed-form expressions are derived for $f^{(k)}$ and $\alpha^{(k)}
(k=1-4)$ involving $1s$ and $2s$ state, for \emph{free H-like ion}.
