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The only property which we could consider to be truly unique to the oxygen atom, meaning that no other atom could ever have that property without also being an oxygen atom, is that it contains 8 protons. Elements are fundamentally defined by the number of protons in their nucleus; any number other than 8 will, by definition, be a different atom than oxygen.
The only other property, one which takes a little more wording in order to make sense, is that oxygen is in a neutral, or "ground state" with 8 electrons arranged in the configuration 1s2, 2s2, 2p4. This means that electrons arranged in this way will counteract the 8 positive charges in the nucleus and create a total charge of 0. It is entirely possible for other elements to have a 1s2,2s2,2p4 configuration, but in their cases these would be ions, because the 8 electrons would have more or less than 8 protons, leading to a different total charge. Another effect of this electron configuration is that it would create a unique series of spectral absorption and emission lines for the oxygen atom, but this isn't something you'd be able to see in a model of the atom.
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