The answer to your question is "no".
Unless you specifically ask to receive paper share certificates, then brokers will hold your shares with a custodian company in the broker's own nominee account. If you are able to receive paper certificates, then the registrar of the company whose shares you own will have a record of your name, however this is exceptionally rare these days. Using a stockbroker means that your shares will be held in the broker's nominee account.
A nominee company is a custodian charged with the safekeeping of investors’ securities. It should be a separate entity from the broker itself.
In essence, the nominee is the legal owner of the securities, while you retain actual ownership as the beneficiary.
Your broker can move and sell the securities on your behalf – and gets to handle all the lovely paperwork – but the assets still belong to you. They can’t be claimed by the broker’s creditors if things get messy.
The main reason for this kind of set-up is cost, and this is why brokers are able to offer relatively low dealing costs to their clients.
You can, if you wish, ask your broker for an account that deals with paper share certificates. However, few brokers will offer such an account and it will mean that you incur much higher dealing costs and may mean that you cannot sell you shares without first submitting the paper certificates back to your stock broker.
Note that the stock exchange plays no role in recording ownership. Nor does your broker's account with the clearing house.