Cid made his debut in earlier FFs than FF7, around FF3/6, 2/4. The reasoning for this, or so I heard was that it was a tribute to someone who used to work on the original FF game or somehow tied into place, by the name of Cid, and died, so they used his name in the FF games afterwards.
Cid, along with being there for advice and what have you, is also in the FF games as the airship pilot or is somehow related to anything that deals with airships in FF, but this may not be true for every single one since FF8 didn't have Cid tied in with airships.
There is a lot of things that may be similar in a FF to other ones, since it's the same creators or some of the same creators, and FF9 definitely shows a lot of FF similarities, but each FF is always different from another one, even though some people like to create theories of another character being a child of previous characters.
All the worlds are different, storylines, etc.
Final Fantasy and connections are vast, so this topic can hold a lot of connections, and a very good FF source full of FF definitions and everything would be FF compendium.