  Studies indicate strong evidence of a scaling relation in the local Universe
between the supermassive black hole mass ($M_\rm{BH}$) and the stellar mass of
their host galaxies ($M_\star$). They even show similar histories across cosmic
times of their differential terms: star formation rate (SFR) and black hole
accretion rate (BHAR). However, a clear picture of this coevolution is far from
being understood. We select an X-ray sample of active galactic nuclei (AGN) up
to $z=2.5$ in the miniJPAS footprint. Their X-ray to infrared spectral energy
distributions (SEDs) have been modeled with CIGALE, constraining the emission
to 68 bands. For a final sample of 308 galaxies, we derive their physical
properties (e.g., $M_\star$, $\rm{SFR}$, $\rm{SFH}$, and $L_\rm{AGN}$). We also
fit their optical spectra for a subsample of 113 sources to estimate the
$M_\rm{BH}$. We calculate the BHAR depending on two radiative efficiency
regimes. We find that the Eddington ratios ($\lambda$) and its popular proxy
($L_\rm{X}$/$M_\star$) have 0.6 dex of difference, and a KS-test indicates that
they come from different distributions. Our sources exhibit a considerable
scatter on the $M_\rm{BH}$-$M_\star$ relation, which can explain the difference
between $\lambda$ and its proxy. We also model three evolution scenarios to
recover the integral properties at $z=0$. Using the SFR and BHAR, we show a
notable diminution in the scattering between $M_\rm{BH}$-$M_\star$. For the
last scenario, we consider the SFH and a simple energy budget for the AGN
accretion, obtaining a relation similar to the local Universe. Our study covers
$\sim 1$ deg$^2$ in the sky and is sensitive to biases in luminosity.
Nevertheless, we show that, for bright sources, the link between SFR and BHAR,
and their decoupling based on an energy limit is the key that leads to the
local $M_\rm{BH}$-$M_\star$ scaling relation.
