Has anyone ever noticed that the lift hill chains on Schwarzkopf coasters, such as the Wildcat at Cedar Point, resemble swing set chains rather than bicycle chains? I wonder what the catch under the car looks like?

From the website:

This chainlift moves up the vehicles on the "Wildcat" and, in combination with another track system, on "Jet Star 1" too. It is a simple link-chain. Every vehicle's bottom features a chain-dog at its rear end, which attaches to the chain while passing it.

http://schwarzkopf.coaster.net/achterbahnen-detail-antriebe-GF.htm

I noticed when I rode Wildcat at Cedar Point that it wasn't like most Chain Lifts, it was an ACTUAL chain, like on a bicycle! It wouldn't rest in the lift, it actually stuck out at the bottom curve of the lift until the car pulled it down. I wonder why it was built like that.

Well, bicycle chains are more similar to most roller coaster chains than the chains used on swings. It all depends on the specifications and design of the ride as to what type of chain is used.