If different drugs have the same receptor as a target, does that imply that they have the same mechanism of action?
It's reasonable to start with an expectation that agonists at the same receptor have the same effect, but that's not necessarily true due to biased agonism. Biased agonism occurs when binding of different agonists to the same receptor results in different downstream activation of second-messenger systems.
Psychedelics and serotonin receptors
A particular example I am familiar with (besides the Jarpe paper) involves the 5HT2a serotonin receptor. This receptor is a shared molecular target of several psychedelic drugs: LSD, psilocybin/psilocin, DMT and others. However, this receptor exhibits biased agonism (Berg et al 1998).
In fact, it seems like agonists at the 5HT2a receptor neatly segregate into those with psychedelic effects (eg: LSD/psilocybin) and without (eg: lisuride/ergotamine, also serotonin itself) based on the different downstream molecular signalling pathways that they trigger, despite binding to the same general site on the same receptor.
Drugs' mechanisms of action can also depend on concentration and on all the other possible targets for those drugs. Pharmaceuticals are often small molecule drugs and while they may have preference for certain receptors, they are rarely (probably never) purely specific.
For example, the NSAID class of anti-inflammatories all work through inhibition of cyclooxygenase enzymes, but they differ in their relative specificity for COX-1 and COX-2. If you are looking at two drugs that have the same effect on COX-2, but don't compare their effects at COX-1, you might be surprised to find they have different results.
For another class, SSRIs used as antidepressants also have effects on other monoamine transporters (dopamine and norepinephrine) but vary in how specific they are.
Berg, K. A., Maayani, S., Goldfarb, J., Scaramellini, C., Leff, P., & Clarke, W. P. (1998). Effector pathway-dependent relative efficacy at serotonin type 2A and 2C receptors: evidence for agonist-directed trafficking of receptor stimulus. Molecular pharmacology, 54(1), 94-104.
González-Maeso, J., Weisstaub, N. V., Zhou, M., Chan, P., Ivic, L., Ang, R., ... & Sealfon, S. C. (2007). Hallucinogens recruit specific cortical 5-HT2A receptor-mediated signaling pathways to affect behavior. Neuron, 53(3), 439-452.
Jarpe, M. B., Knall, C., Mitchell, F. M., Buhl, A. M., Duzic, E., & Johnson, G. L. (1998). [D-Arg1, D-Phe5, D-Trp7, 9, Leu11] Substance P acts as a biased agonist toward neuropeptide and chemokine receptors. Journal of Biological Chemistry, 273(5), 3097-3104.