Bipolaron states in ellipsoidal quantum dots are examined within the framework of Feynman's path-integral formalism. The following characteristics of these quasiparticles are calculated: the binding energy, the number of phonons in the bipolaron cloud and the radii. The impact of both shape and size of a mesoscopic structure on the bipolaron characteristics is studied. It is shown, that in the case of strong confinement the influence of the confinement geometry is very pronounced. Two different cases are possible. (i) When the radius R_k, corresponding to a given k-axis of the ellipsoidal potential well is fixed to be less than the radius R₀, at which the bipolaron binding energy in a spherical quantum dot achieves its maximum, the bipolaron binding energy passes consecutively through a local maximum and a local minimum with increasing the confinement strength corresponding to the other axes (i ≠ k). (ii) When R_k ≫ R₀, the bipolaron binding energy shows only a monotonous growth with decreasing the effective sizes of the structure along the other axes (i ≠ k).