Connection request keeps popping up

I can't be positive, but I believe this is due to your Macs trying to coordinate the information via your iCloud connection. I think you either need to follow the instructions or not have both computers listed in the same iCloud account. Incidentally, having both on the same account means that what you do in one computer will change what is in the other computer when they share the same iCloud account. I don't think that is what you want.

That's a Bluetooth connection request. On each device, open Bluetooth Preferences (System Preferences).

Look at the devices listed there. Find the other computers that are trying to connect to each other and remove each from the others Bluetooth Preferences. This is done by selecting the x that appears when you move the cursor to that device in the list.

I really understand your frustration. Personally, I started reading things on this forum to learn more. I have used Macs for many years and am fairly knowledgeable, but there is so much more to know. Personally I have observed some things on this forum which I will share with you. My opinions only. I see a lot of problems with Apple's Mail app. I stopped using it years ago because I didn't like it. I like Thunderbird. Also, the same with Time Machine which I also tried and didn't like. It is a good app, but not for me. Lots of reports of problems. When doing downloads of updates or upgrades, always use an ethernet connection directly to the modem/router to avoid corruption. People think iCloud is for back up. It is not. It is for transfer data to all associated devices. You said "Changing contents of one computer doesn't mean change on another". I have experienced that it did. My wife erased some Safari Bookmarks on one computer and the other one lost its Bookmarks and duplicated the other's, so I disconnected them. I don't really need or use iCloud anyway. I like to keep things as simple as possible.

Lastly, I see recurring problems with people who wait forever to do updates and upgrades and then have all kinds of problems. My devices are always up to date and have few if any problems. I don't install junk apps. I do regular maintenance like restarting my computer, occasionally doing a restart in the Safe Mode and clearing Browsing data. I have an external disk which is fully backed up and has its own operating system on it. My files are always backed up to the minute. I am fortunate that I know how to troubleshoot from my many years in telecommunications. It's mostly a matter of isolating, trial and elimination. So, hang in there, my friend. The world is difficult and complex. We have to evolve, like it or not.

Check your Sharing Preference Pane and turn off Bluetooth Sharing.

Turn off Bluetooth off on all devices. Exit the Bluetooth Preference Pane if it is open. Then on all devices, delete this file:

/Library/Preferences/com.apple.Bluetooth.plist

If it is also present in your user libraries, delete it there also.

You can restart all devices at that point and see if the issue is gone (turning bluetooth back on where you need to). If it were me, before restarting I would boot each device into safe mode in order to clear out caches that might "remember" some problematic setting. Then restart normally and see if the issue is gone.

Other resources are available to you. Hold option and open the Bluetooth icon in the menubar, you will see more info than without holding the option key. Also holding SHIFT-OPTION when opening the Bluetooth icon in the menubar presents you with some DEBUG options which you can experiment with. You may need to do this on multiple computers.

If Bluetooth Sharing is OFF (as shown in your picture) for all your devices, then that's not causing the problem, clearly.

There is an additional thing you can try after Safe Mode, which would be to reset NVRAM, but I think that's a long shot because the items stored there do not (to my knowledge) pertain to Bluetooth. But it would not hurt to try if running out of options.

Deleting Preference files is an old and standard troubleshooting step that sometimes resolves issues where a setting is not behaving as the user expects or requests. It should not be necessary, but these files can be damaged when there is a computer freeze, unexpected restart, forced shutdown, power failure, something of that ilk. It's hard to make everything bulletproof. I have had very good luck with various iCloud options across 4 different computers and one iPhone, and all have Bluetooth on (for wireless keyboard, mouse, headphones, Fitbit etc.) but I have never seen that problematic pop up. There might be a connection to iCloud if some stuck or corrupted setting is being propagated across your computers via iCloud, so if you are going to try what I suggested earlier, you might want to turn off iCloud sharing (temporarily) before embarking on that.

Piotr Kowal wrote:

Honestly, I think Microsoft OneDrive is much much better in that. I should have bought Windows machines instead of this crap that overpromises and underdelivers. The only good product they have is iphone.

Restarting in safe mode - seriously? Why would those advanced... sorry - amazing computers don't do it themselves?

If you're having trouble with iCloud for file sharing, you could try other options. I use Dropbox, for instance, it works across various computers running 10.11 to 10.15 as well as the iPhone. Google also has cloud file sharing capabilities you can choose to use.

Sorry to suggest Safe Mode, but what it does is (a) check the disk/file system to make sure it is clean, and (b) clear some caches. The caches might be persisting in reasserting a setting that you have been trying to change. I am suggesting these things because we are trying to troubleshoot your problem. The things I have suggested are very basic and simple to do. I and others I know have had good experiences with iCloud but I don't doubt the reality of what you have been reporting and experiencing.