I assumed Gary Seven was one of the descendants of the human stock taken by the aliens six thousand years before.
That's exactly what was intended, as the dialogue itself indicates. Actually it's a little ambiguous; Gary only says that "agents" are descended from those abducted humans, and he's a supervisor, not an agent. But the pitch document that was written a couple of weeks ahead of the episode script clearly states that he is one of those descendants.
I also assumed—without foundation—that he was the seventh man named Gary in their program, and so Gary VII, actually.
"Gary Seven" is his code name, according to dialogue. Presumably it's not his real name. After all, would the name "Gary" have existed 6,000 years ago when his ancestors were taken from Earth? (Its ancestor, the Gaelic name Garaidh, has only been traced back as far as the Middle Ages, sez Wikipedia.) I always figured that, since the people who gave him the code name were aliens, they were a little bit off the mark in their attempt to coin a plausible human/Western name. Although in Watching the Clock
I referred to other Aegis agents as Cyral Nine and Rodal Eight; I decided at the time to treat them as ranks or tiers, with lower numbers being higher echelons. I've sometimes wondered if that was really a good idea.