It appears to be a feminine adjectival suffix, which is/was apparently also used as a patronymic suffix:
https://en.wiktionary.org/wiki/-ene#English
Another such suffix is
-ine, of which the masculine equivalent is
-inos (cognate to Latin
-ina and
-inus). So no,
-one is not the only Greek suffix of this type.
Forchta in biuonga quamon ouer mi, in bethecoda mi thuisternussi.
In ic quad: "uuie sal geuan mi fetheron also duuon, in ic fliugon sal in raston sal?"