Patchy colloidal platelets with non-spherical shapes have been realized with different materials at length scales ranging from nanometers to microns. While the assembly of these hard shapes tends to maximize edge-to-edge contacts, as soon as a directional attraction is added-by means of, e.g. specific ligands along the particle edges-a competition between shape and bonding anisotropy sets in, giving rise to a complex assembly scenario. Here we focus on a two-dimensional system of patchy rhombi, i.e. colloidal platelets with a regular rhombic shape decorated with bonding sites along their perimeter. Specifically, we consider rhombi with two patches, placed on either opposite or adjacent edges. While for the first particle class only chains can form, for the latter we observe the emergence of either chains or loops, depending on the system parameters. According to the patch positioning-classified in terms of different configurations, topologies and distances from the edge center-we are able to characterize the emerging chain-like assemblies in terms of length, packing abilities, flexibility properties and nematic ordering.