See a German definition: Geonyme sind geographische Begriffe, die als Pseudonyme verwendet werden. Z. B. der Name eines Ortes (meist des Geburtsortes!), eines Flusses etc.
Gerade im 15. und 16. Jh. war es wohl besonders hipp, den Namen des Geburtsortes für ein solches Pseudonym zu wählen. Siehe Liste.
Gives examples such as Offenbach, born in Offenbach as Josef Ebst. Wikipedia in English inaccurately says "born Joseph Offenbach". Hieronymus Bosch was born van AKen, but adopted the name of his birthplace, Hertogenbosch.
Since the word geonym is so uncommon in English, we had perhaps better adopt the meaning given in a language where it is used.
I rather think that geonym could be a surname or name derived from the name of a particular place. For example, in Hard Times, there is Stephen Blackpool. In not English languages it seems common - Persian is full of names like Mashadi, Teherani, Tabrizi.
I found it used to mean "country name." Does this fit with the definition "geographical feature"? I doubt it. But geographical features are: glaciers, contour lines, deserts and so on. They are common nouns.