There's many possible reasons why Doomfist III is depicted with Omnic scripture. Here's 2 that I can think of from the top of my head:
The usage of language/script also reflects on the history of the country of Numbani. By using the Latin alphabet, the museum makes a statement that Doomfist I and II were before the time that Numbani embraced the union between Mankind and Omnics.
Or Doomfist III himself wants his name to be written in Omnic for whatever reason. That might be because he is an Omnic of course, but it also could be because he wants to represent that harmony between both peoples (being a cyborg would emphasize this as well, half man half omnic), or because he grew up on the Omnic alphabet or identifies himself more with Omnic culture because, for instance, he was an orphan who was adopted by Omnic parents or something like that.
That Omnic script, while heavily hinting at Doomfist III being an Omnic, does not necessarily prove that he is. However, we can surely consider him to be someone who wants to fight for Omnic rights and equality, regardless of what people he belongs to.