We study the entanglement properties of the shape phase transitions between different geometric limits of
two coupled equivalent molecular benders modeled with the two-dimensional limit of the vibron model. This
system has four possible geometric configurations: linear, cis-bent, trans-bent, and nonplanar. We show how the
entanglement, accessed through the calculation of the linear entropy, between benders and between rotational
and vibrational degrees of freedom changes sensitively in the critical regions of this two-fluid bosonic model.
The numeric calculation is complemented with a variational approach to the ground-state wave function in terms
of symmetry-adapted coherent states.