Why is the Planck length considered fundamental, but not the Planck mass?

From the perspective of particle physics, you are correct, the Planck length and Planck mass are essentially equivalent concepts: the Planck mass describes a (very high) energy scale ($sim 10^{19}$ GeV) at which new physics must emerge, just as the Planck length entails a (very short) length scale beyond which we need a new description. If we set $hbar=c=1$ (which are really just conversion factors between units) we see that they are inverses of each other, $m_P=1/l_P$.

More precisely, if we take the Einstein-Hilbert action for gravity and expand around a flat metric $g_{munu} = eta_{munu} + h_{munu}$, where we can interpret $h_{munu}$ as the graviton field, the resulting action will have an infinite number of higher order terms suppressed by powers of the Planck mass. Roughly, we have $$mathcal{L}_{EH} sim frac{1}{2} partial hpartial h+ frac{1}{m_P} hpartial h partial h + frac{1}{m_P^2} h^2partial h partial h + ldots $$ (as well as terms from higher derivative corrections, which are also higher order in $1/m_P$). So we have predictive control at energies scales much less than $m_P$, where the infinite number of higher order terms can be ignored. But once we reach the Planck scale (i.e. energy scales of $m_P$ or length scales of $l_P$) the non-renormalizable effects become important and all the quantum corrections and higher order terms render the above Lagrangian equation useless, and we require a new description.