Website says enable cookies even though they are enabled, on Catalina I use the bill pay feature from my credit union.  After upgrading to Catalina the bill pay feature reports that cookies are disabled.   

Cookies are not disabled.  I have cleared them, cleared history, restarted safari, restarted the computer.  Still no love.  You can see that new cookies have been created after logging in to the credit union.  But the bill pay feature still reports that cookies are disabled.   I have two systems running Mojave that don't have this problem.  What could be blocking this web site from reading the cookie? The firewall maybe?
If there's any hint of "best viewed in these browsers" on the site, it's good to use them. 

You can "empty cache" using alt+command+E or by using Develop Menu in Safari.

Also, try turning off "Prevent cross-site tracking" to see any effect. You can turn it back on later, if you want.


  Some websites use third-party content providers. A third-party content provider can track you across websites to advertise products and services.
  
  With this option turned on, tracking data is periodically deleted unless you visit the third-party content provider.


https://support.apple.com/en-in/guide/safari/sfri35610/12.0/mac/10.14.5

Also disable all extensions and third party safety applications you might have.