Recently superconductivity at the interface between the insulators LaAlO3 and SrTiO3 has been tuned with the electric-field effect to an unprecedented range of transition temperatures. Here we perform a detailed finite-size scaling analysis to explore the compatibility of the phase-transition line with Berezinskii-Kosterlitz-Thouless (BKT) behavior and a two-dimensional-quantum-phase (2D-QP) transition. In an intermediate regime, limited by a gate voltage dependent limiting length, we uncover remarkable consistency with a BKT-critical line ending at a metallic quantum critical point, separating a weakly localized insulator from the superconducting phase. Our estimates for the critical exponents of the 2D-QP-transition, z similar or equal to 1 and nu over bar similar or equal to 2/3, suggest that it belongs to the three-dimensional-xy universality class.