Emulsion droplets are a versatile system whose interactions can be tuned to mimic cellular functions, such as domain formation, adhesion or hemifusion. For example, immiscible lipids on the surface of emulsion droplets create stable patterns of circular or stripy domains, reminiscent of lipid rafts in cell membranes. Functionalizing the lipids with biotins allows them to bind to each other either irreversibly through streptavidin or reversibly through complementary DNA strands. We show that these mobile adhesion patches self-assemble linear chains of thermal droplets into compact structures, such as flowers. The size of adhesion is predicted by the balance between the binding energy and the energy of deformation of the droplets. Applying an external pressure to the system strengthens adhesions, which highlights the importance of homeostatic pressure on cell-cell adhesion and tissue integrity in vivo. Alternatively, functionalizing the lipids with E-cadherin proteins unexpectedly leads to droplet kissing and fusing together up to a given droplet size. Microscopically, we find that the lateral cis-interaction of the proteins clusters them into rings upon adhesion, which in turn ruptures holes in the lipid monolayer to cause fusion. Therefore biomimetic emulsions provide not only a new class of liquid patchy particles for self-assembly, but also a rich playground to address important problems in biophysics.