Installing the Safari upgrade, my cookies are blocked. Cookie settings are unblocked.

If Safari preferences are set to allow cookies, then cookies will (should) be allowed.

If there a particular web site access issue, or error message, or other access issue that you’re seeking to address?

Boot into Safe Mode, and check whether the browsing works as expected there; that rebuilds various caches, and disables various add-ons.

Cross-site and related web site advertiser shenanigans will be blocked.

Also check for and remove remove the add-on anti-malware apps should any of that have bene installed.

You’re using the Safari web browser. You have cookies enabled in settings. You are getting errors from two financial institutions, reporting problems saving cookies. You believe this is related to a security update.

Have you tried booting through Safe Mode? Safe Mode rebuilds parts of the boot-time environment, and this rebuild can sometimes resolve patch-related errors.

Do you have add-on anti-malware apps, add-on VPN clients, add-on cleaners, or related apps, installed? These have been known to intercept communications, and to cause various problems with secure connections.

I did not suggest turning off the add-on anti-malware and/or add-on cleaners and/or VPN clients, and ilk.

I'd suggested removing the anti-malware and the cleaner apps per the vendors' instructions, rebooting, and try again.

I'd also remove any Safari browser extensions.

The anti-malware apps are effectively user-installed malware that doesn't like to share, in how they tie into a system. Both the malware and the anti-malware harvests data, too. Add-on anti-malware and cleaner apps are also very common sources of corruptions, crashes, hangs, and other messes. Safari extensions can also get involved in creating messes here, too.

With some of these add-on apps, it wouldn't be surprising that there'll be the need to remove the apps entirely as suggested above, and then back up the system to external storage once or twice, boot Recovery or boot a bootable installer, wipe, and reinstall macOS, and migrate in the content. Some of these add-on apps hose the environment, and this environment looks hosed. This re-install to un-do the damage that some of these apps have caused, and then migrate in your content and your (not-security-related) apps.

Various of the add-on anti-malware apps are by all appearances little more than scams, and widely advertised cleaner apps have been on the wrong end of lawsuits. Yes, there are some few of these apps with decent reputations around the forums. The rest? Not so much.