What is the rationale for calling something as an 'inorganic [organic compound]'?
I would not call graphite organic, but there is no clear-cut way of defining organic and inorganic compounds.
To your question, the pairs of compounds are isoelectronic. That means that if we assume that molecular orbitals (MOs) arise from atomic orbitals, the corresponding MOs in the two compounds have the same occupancy. In these cases it arises because the number of electrons in the dimer BN is the same as in the dimer CC. Another example of isoelectronic compounds would be N$_2$ and CO. Being isoelectronic, they do share some properties, e.g. both N$_2$ and CO have a triple bond. If only the valence electrons are identically occupied, we call them isovalent, which would include e.g. H$_2$O and H$_2$S.
The bonding of benzene and borazine differs in that not all sites are equivalent, but the form of the MOs is broadly similar and the same applies to boron nitride. Similarity in MOs is then reflected in similarity in reactivity. (Even though they are not that similar in reactivity.)