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there is more to organic compounds than just containing carbon. connectivity of atoms also plays an important role.
some chem textbooks assert the fact that organic compounds should also contain at least 2 C-H bonds. meanwhile, the hydrogen in H2CO3 is bonded to the oxygen, rendering the molecule, according to this definition, inorganic.
but what shall be done in the exceptional case of mellitic anhydride, the largest oxide of carbon which contains no C-H bonding? it is generally accepted that, by originating from an officially accepted organic molecule (in this case, mellitic acid), another molecule (the product of that starting molecule) can be placed in the same category. but this definition is ambiguous as well because certain chemical, more intricate chemical pathways may generate inorganic products from an organic substrate.
that's what i'd say it is borderline. it is organic enough to be embedded in biochemical processes, while also being of interest for inorganic chemists because of its features (lack of stability, connectivity, acidity, there are even some dissertations which treat carbonic acid from a physical point of view, associating it with other inorganic acids).
