When you own bitcoin, what you own is a key that gives you access to a bitcoin credit on the bitcoin blockchain thats assigned to your private key. The key allows you to transfer some or all of your credit directly to another key.
Technically that is not what happens. But for a high level explanation, it is okay.
When it's turned into fiat, what does the blockchain record? Not that ownership is transferred to a different private key. Does it record an ID of the exchange that converts the currency?
What happens is the exact same thing that happens when you trade Bitcoin (or any currency) for any good (e.g a fiat currency). You transact some Bitcoin to another person who transactions some fiat to you. Blockchain records nothing except that Bitcoin went from one address to another. Only those involved in that trade will know that another transaction happened in another currency to transfer some fiat from the other guy to you.