I would like to allow 3rd party cookies

If you don't check the box to block cookies, you should get 3rd party cookies. If you are not getting all the expected/required cookies, in Safari/Preferences/Privacy, try unchecking Prevent cross-site tracking to see if that helps.

Are you running any anti - virus programs? Using a VPN?

Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin

Etrecheck – System Information

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

Apple doesn’t routinely monitor the discussions. These are mostly user to user discussions.

If you want to, you can send Apple feedback. They won't answer, but at least will know there is a problem or a suggestion for change. If enough people send feedback, it may get the problem/suggested change solved sooner.

Safari Feedback

One of my financial websites insists Safari is blocking cookies when the Privacy permissions are set to not block cookies. I was relieved to see this is apparently a bug Apple has been slow to fix in 13.0.4. The Firefox browser allowed me to access what I needed but I am disappointed Apple hasn't yet addressed this issue as it's a really big problem if Apple wants people to continue to use Safari. It seems whenever a Safari problem crops up, most businesses immediately suggest I switch to Chrome. Seems Apple would not want to drive people to Chrome but that's what is happening. Most of my friends use Chrome but I'd rather not.

I have had this same problem, my bank website keeps telling me I need to not block cookies, but I have tried several steps: checking block then unchecking to enable, and then quitting & restarting Safari. This did not work. Then I tried again to allow cookies in Preferences, closing Preference window to save, clearing History in Safari, quitting Safari then restarting. Still does not work. Please fix this!!

I too have had to go to chrome because my bank has said the same thing to me. I seen many hours with Apple Help and they too could not resolve the "cookie" issue. I'm extremely disappointed that Apple is not interested in resolving this major problem.

I tried this and it still blocks the cookies needed for a certain site I need to use.

Safari/Preferences/Privacy.

Also see the last post in this discussion to see if it helps.

Safari version 13.0.4 enable cookies - Apple Community

I have the same problem. iPhone 8plus iOS 13.1.3, a real pain. I have done all the steps recommended several times but problem persists. I can’t be you only one. Help

I am also facing the same issue, I have tried in safe mode, other user account, without vpn and without anti-virus

Still getting the same problem

Its happening in latest mac os catalina

I am unable to use cloud9 & some others as third party cookies have been removed from safari 13. Is there any work around that anyone knows.

Worked fine in signing in to Discovery and A&E before the last update. Signing in with Chrome no issue.

My bank is telling me that I must Stop preventing cross site tracking also Stop website tracking and to not block cookies from there site. How do I do this???