I know nothing about the world of crypto currencies, but I do know finance.

The exchange does not hold the trades instruments/currencies/securities as assets. The business of a normal exchange is normally risk free (just matching buyers to sellers). Some exchanges step in as middle man in the trades, a process that I believe is called novation of the trade. The original trade between the buyer and the seller is novated, transformed into two trades, both against the exchange, one for each party and opposite direction. In this case the main risk is counterparty risk, the risk that one of the counterparties fail in some way.

An exchange never holds its own positions.

What is the difference in these cases? Have the crypto exchanges somehow used their users cryptosecurities as assets?