Cookies on safari

Go step by step and test.

Step.1 Update Safari.

Update or reinstall Safari for your computer - Apple Support

Step.2 Try iCloud syncing

Check your account details 

Choose Apple menu , and click iCloud. Make sure that you're signed in with the same Apple ID that you're using on your other supported devices.

Turn off iCloud Bookmarks and turn it back on

1. Quit Safari.
2. Choose Apple menu  > System Preferences. If you're using macOS Catalina, click Apple ID. Click iCloud.
3. Deselect Safari.
4. Close System Preferences and wait about a minute.
5. Open System Preferences and click iCloud.
6. Select Safari.
7. Open Safari.

Restart your computer

Restart your computer and see if the issue still exists.

https://support.apple.com/en-us/HT203519#OSX

Step.3 Try Safari Technology browser

Download Safari Technology Preview and install it.

This version of Safari has advanced features’

Safari Technology Preview  Download Safari Technology Preview

Safari Technology Preview > Preferences >  Privacy

UnCheck the boxes beside “Prevent cross-site tracking” and “Block all cookies”.

Safari > Preferences > Websites > Pop-ups

Choose “Allow” in the box next to website’s address.

Test.

Are the cookies being blocked? Others are seeing the same thing. It seems to be a bug in the newest Safari 13 version.

Block cookies is off but being limited by… - Apple Community

Try another browser until the problem is fixed.

Firefox        10.9 or higher

Opera Browser        10.9 or higher

SeaMonkey      10.9 or higher8

Waterfox          10.8 or higher