Why Are Final Fantasy VII Characters in Ehrgeiz?
Start with who made what. Ehrgeiz was developed by DreamFactory under Square, in a joint venture with Namco. Final Fantasy VII was Square's own game, released in Japan in January 1997 to enormous success. When the Ehrgeiz team wanted guest fighters, the most famous RPG cast in the world was an internal request, not a licensing negotiation.
Then look at the credits. The artist of Ehrgeiz, Tetsuya Nomura, designed the game's original characters and its Final Fantasy VII characters alike. Cloud, Tifa and Sephiroth did not arrive as licensed likenesses. They arrived drawn by the same hand that had created them, which is why they sit inside this game's art style instead of on top of it.
THE TWO COVERS
Same game, two shelves

WHO IS IN, AND WHERE
Two in the arcade, six at homeTHE DOCUMENTED LINKS
Why it was possible, and why it made senseTHE HONEST FOOTNOTE
What is documented, and what is notOne caution belongs on the record. No creator interview in the sources for this page gives a single official sentence of motive, of the form: we added Cloud because of this. What is documented is the structure around the decision: same publisher, same character designer, a reveal schedule in the arcade that treated the guests as events, and contemporary press that understood the characters as the game's draw.
So the answer stands on those documents rather than on legend. Final Fantasy VII characters are in Ehrgeiz because Square could put them there with nobody's permission but its own, because Nomura's presence made them native rather than pasted in, and because in 1998 there was no faster way to tell RPG players that a fighting game was also for them.