DeletedUser125 So if I transfer coins from a regulated broker to a wallet, the wallet then gets marked with my ID?
The broker knows who you are, and they know the address that the funds were transferred to. They may be legally required to share this information with tax or other financial authorities. They will certainly retain this information in their records, which might also get leaked in the future.
In the case of Bitcoin, that address is on a public ledger, and literally anyone can see the address, how much it holds, the address it came from, and any addresses it is sent to in the future.
The only thing not public is who owns that address. You are really trusting the broker, and anyone they share that data with to keep that information private.