I address this topic in my books but the short answer is god impersonation exists.
Can you always tell the real from the impersonators? No. Often the impersonators have even more money, power, and resources than the embodiments because there are humans actively devoted to discrediting, destroying, or replacing the gods.
That is not to say humans always call gods by the same names. Over time, gods (as concept and embodiment) do migrate to different lands and cultures.