We study the impact that large-scale perturbations of (i) the matter density and (ii) the primordial gravitational potential with local primordial non-Gaussianity (PNG) have on galaxy formation using the IllustrisTNG model. We focus on the linear galaxy bias $b_1$ and the coefficient $b_phi$ of the scale-dependent bias induced by PNG, which describe the response of galaxy number counts to these two types of perturbations, respectively. We perform our study using separate universe simulations, in which the effect of the perturbations is mimicked by changes to the cosmological parameters: modified cosmic matter density for $b_1$ and modified amplitude $mathcal{A}_s$ of the primordial scalar power spectrum for $b_phi$. We find that the widely used universality relation $b_phi = 2delta_c(b_1 - 1)$ is a poor description of the bias of haloes and galaxies selected by stellar mass $M_*$, which is instead described better by $b_phi(M_*) = 2delta_c(b_1(M_*) - p)$ with $p in [0.4, 0.7]$. This is explained by the different impact that matter overdensities and local PNG have on the median stellar-to-halo-mass relation. A simple model of this impact allows us to describe the stellar mass dependence of $b_1$ and $b_phi$ fairly well. Our results also show a nontrivial relation between $b_1$ and $b_phi$ for galaxies selected by color and black hole mass accretion rate. Our results provide refined priors on $b_phi$ for local PNG constraints and forecasts using galaxy clustering. Given that the widely used universality relation underpredicts $b_phi(M_*)$, existing analyses may underestimate the true constraining power on local PNG.
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