The "President Shinra" thing isn't that weird to me, because President Shinra is very obviously the effective head of state of Migdar, a kind of "Corporate Monarch," and that's just how monarch names work.
It's most obvious with Japanese Emperors, what with FF7 being a Japanese game: Naruhito is Emperor of Japan and belongs to the Yamato Dynasty, but he's not "Naruhito Yamato." He's just Naruhito (although to the Japanese press he is currently specifically "His Majesty the Emperor" and never referred to by name).
The British monarchy is different, I think? "Mountbatten-Windsor" is their surname but you don't use it as a surname (unless you're the French press, who insist on referring to British monarchs as "Elizabeth Windsor" or "Charles Windsor.") Charles III in normal nomenclature is just "King Charles."
So to me it parses as totally coherent for President Shinra to be using "Shinra" as his "royal name" - if he had a surname before, it got erased from the historical record. He's just "Shinra," and by extention under his full honorific title, "President Shinra."
The only thing that makes that weird is Rufus. Because Rufus is Rufus Shinra instead of, like, Shinra II. It'd be like if Charles ascended to the throne and took on the name Charles Elizabeth. He throws a wrench in a very simple analogy.
It's weird to say but out of all the FF7R characters who got a glowup from the original, President Shinra is absolutely one. He doesn't get to shine much but his final confrontation with Barret actually had some charisma even though he starts the scene cringing and begging for his life.