The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of "equal temperature" (ET), or "in equilibrium" (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can he in equilibrium and yet have different effective temperatures. A recent result [J.-T. Hsiang and B. L. Hu, Phys. Rev. D 103, 065001 (2021) for Gaussian quantum systems shows that an effective temperature r can be defined at all times during a system s nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system s T* is slightly higher than the bath temperature T-B, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature T-B will not have the same equilibrated temperature as the first object. Thus ET not equal EQ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath B, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that "temperature" appearing in thermodynamic relations becomes universally defined and makes better physical sense.