A little more nuanced take: the transfer agent maintains a list of shareholders at the company's behest, the DTCC is one step further down the chain, operating as a clearinghouse between brokers and banks. The list of shareholders is managed by the company or it's bookkeeper, the "transfer agent". Insiders and people who own paper certificates are on the list kept by the transfer agent, directly owning their shares on the company's books, alongside Cede & Co (a partner of the DTCC), who owns the bulk of all publicly traded shares. Cede & Co, and it's partner, the DTCC, then maintain a list of which shares beneficially belong to each of the DTCC members (brokerages, banks). The brokerages and banks in turn maintain lists of which clients beneficially own the brokerage's shares.
Yeah for every explanation there’s another more complicated one, it’s a pretty crazy mess. A long chain of abstractions and not every one uses the same stack.
For one commonly used transfer agent's take, see: https://www.computershare.com/us/becoming-a-registered-share...