I will give you my "understanding" of how this does not contradict what is known about photons at the elementary particle level.
The publication here describes a complicated system of guiding photons through a special medium
We search for a photonic trimer using an ultracold atomic gas as a quantum nonlinear medium. This medium is experimentally realized by coupling photons to highly excited atomic Rydberg states by means of electromagnetically induced transparency (EIT)
So this is a system with which photons interact and display three photon correlations which induce a mass on the photon following the mathematical model that fits the data:
is the effective photon mass, $a$ is the scattering length, $Ω_c$ is the control laser Rabi frequency, and $∆$ is the one-photon detuning.
So it is within a complicated mathematical model that the photons acquire mass.
To get a perspective on this I think of the $π^0$, it decays into two photons with an angle between them , and this is defined by the mass of the pion. The four vectors of the two photons added will still have the mass of the $π^0$ , the momentum and energy .If one timed the photons, it would seem that they moved with a velocity less than c, but in effect they traverse a longer path than the $π^0$ path, because of the angle between the two photons. This helps me realize that sums of four vectors are not intuitive. The four vectors involved in these photonic states are more complicated, as they are defined by interactions with a system of particles ( the cold gas). This ties up with this popularized explanation:
So why are the normally lone-ranging photons suddenly interacting with each other? The team's hypothesis is that as photons bump into the rubidium atoms, they form polaritons – quantum particles that are part-light and part-matter. Polaritons have mass, which is how they can bind to other polaritons. Once they leave the cloud, the atoms they've picked up stay behind, but the photons remain bound together.
They are not bound, they are correlated, the way the two photons of the $π^0$ are correlated.