We use hydrodynamical separate universe simulations with the IllustrisTNG model to predict the local primordial non-Gaussianity (PNG) bias parameters $b_{phi}$ and $b_{phidelta}$, which enter at leading order in the galaxy power spectrum and bispectrum. This is the first time that $b_{phidelta}$ is measured from either gravity-only or galaxy formation simulations. For dark matter halos, the popular assumption of universality overpredicts the $b_{phidelta}(b_1)$ relation in the range $1 lesssim b_1 lesssim 3$ by up to $Delta b_{phidelta} sim 3$ ($b_1$ is the linear density bias). The adequacy of the universality relation is worse for the simulated galaxies, with the relations $b_{phi}(b_1)$ and $b_{phidelta}(b_1)$ being generically redshift-dependent and very sensitive to how galaxies are selected (we test total, stellar and black hole mass, black hole mass accretion rate and color). The uncertainties on $b_{phi}$ and $b_{phidelta}$ have a direct, often overlooked impact on the constraints of the local PNG parameter $f_{rm NL}$, which we study and discuss. For a survey with $V = 100{rm Gpc}^3/h^3$ at $z=1$, uncertainties $Delta b_{phi} lesssim 1$ and $Delta b_{phidelta} lesssim 5$ around values close to the fiducial can yield relatively unbiased constraints on $f_{rm NL}$ using power spectrum and bispectrum data. We also show why priors on galaxy bias are useful even in analyses that fit for products $f_{rm NL} b_{phi}$ and $f_{rm NL} b_{phidelta}$. The strategies we discuss to deal with galaxy bias uncertainties can be straightforwardly implemented in existing $f_{rm NL}$ constraint analyses (we provide fits for some of the bias relations). Our results motivate more works with galaxy formation simulations to refine our understanding of $b_{phi}$ and $b_{phidelta}$ towards improved constraints on $f_{rm NL}$.
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