When my wife gave birth to our son at the hospital, I have to put down my phone number as part of the check in form.
Immediately the next day I got call for "Home care services for new mom and baby".
Oh totally. But they don't sync that information "immediately". Nor would they ever want to because then the user would know that's where the information came from.
I guess it depends on what OP meant by "immediately". If they meant the same day, maybe. If they meant within seconds or a few minutes, which is what I interpreted it as, then probably not. It takes time to transfer data out of a secure network, unless they gave the company direct access to a feed from the website, which would be really risky for a bank to give any organization a direct, real-time feed of any kind that is on the same network as financial data. I mean unless the bank also owned the spamming company, but that seems risky for reputation.