The answer to this question depends on the reason why you are using a VPN in the first place. From a privacy perspective, VPN's serve two purposes:
VPN's hide from your ISP (or your local network admin) the sites that you are visiting. Your ISP only sees your connection to the VPN, and not the domains or the IP addresses of the sites that you visit.
VPN's hide your 'real' IP address from the sites that you visit. The sites that you visit only see your VPN's IP address, not your real IP address.
If you are concerned about your ISP seeing that you are visiting your bank's site, then use a VPN.
But, I suspect that the advice that you are referencing has more to do with (2) above. Some sites attempt to fingerprint their visitors (for tracking and data collection purposes), and visitors' IP addresses can be used as part of a fingerprint. This is especially true nowadays, where even dynamically assigned residential IP addresses can remain the same for months or years. But, if it's a site that you login to (such as a banking website), then this is a moot point, because the site knows exactly who you are the moment login (regardless of your IP address).
Finally, with regard to a site triggering a second authentication factor when your IP address changes - I'm not sure how this relates to your underlying question about privacy. Yes, if the site is setup to require a second authentication factor - then using the site without a VPN, then with a VPN will trigger this. But, so would logging into your banking website from your home without a VPN, then from the wifi at your local coffee shop without a VPN later that same day.