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This probably refers to the peculiar Polish custom of electing kings. The electorate comprised a privileged caste known as the szlachta, sometimes translated as nobility but not the same. There were hundreds of szlachta families whose members had the right to participate in the process of choosing a ruler from amongst themselves, so the right was not particularly exclusive.
All that being said, I've never seen a list of szlachta families that included the name Tarkowski; Targowski, however is a szlachcic surname.