Is there any on-block exchanges? I'm new to bitcoin and just learned that most (or maybe all) of the exchanges are working off-block, meaning that they don't really push data to blockchain most of the time.

Is there any altcoins or exchanges that store data in blockchains ? I'm doing researched project on the matter and trying to find exchange that have some of it's trading public.

As I understood from this outdated article, Kraken might been on-block exchange in the past.

BitSquare might be something I'm looking for, but I'm not sure exactly how does it work.
Generally speaking, decentralized markets such as LocalBitcoins and BitSquare have the money onchain as they facilitate exchanges directly between users, and centralized markets that take custody of the money prior to trading have the funds offchain as otherwise the transaction fees they're spending would be much higher.

I think that in the past Coinbase and Kraken both kept user deposits in separate addresses for each customer.