To be centralized means to trust somebody else to handle your money.
In the past, the word “centralized” was a given for all institutions that managed finances.
To be centralized means that there is a trusted middleman to handle whatever asset may be in a trade. In a bank, for example, a customer gives their money over to the bank to hold for them. This one institution is now in complete control of the customer’s money.
In many cases, this is much safer than a person finding some way to manage themselves. Banks have many securities and a team to watch over their customers’ money. The bank can also offer a variety of services, such as loans, because the bank has a large amount of money and has created a trust relationship with the customer.
Centralized cryptocurrency exchanges are no different. A user can store their money on the exchange. The currency is now in the hands of the exchange, but the trust of the middleman makes it easy for a customer to recover a lost password or 2FA because that customer has given the exchange full access to their account. This can also take the pressure off of the customer of being 100% in control of their money. There are many stories of investors losing hundreds of thousands of dollars because they lost the private keys to their hardware wallet. If their money were in a centralized exchange, they wouldn’t have to worry about that; recovering would be as easy as showing a passport or verifying identification.