Welcome to DEVONthink and the forum.
Please search the forum for that error messages and you’ll see that for some people (many who report here) how this is a message from Apple’s servers. For some, Apple’s servers have been unreliable for months. You might try Apple CloudKit, if you really need a third party internet synch service. Otherwise, also or instead of, try Bonjour. More reliable and quicker.
I can do without iCloud for now, but probably not in the long run. I disabled it for now. Bonjour is enabled on the Macbook. But on iPad the cloud button is completely greyed out (see screen print above), so no idea how I can activate bonjour there. Any clues would be appreciated…
Please search in the forum first. There are a gazillion of posts on sync, there’s a chapter in the manual and afaik there’s also a blog post on the DT site. No need to repeat all that.
Yes there are a gazillion of posts with problems. Looking for the one with the solution. So far havent found it…
There is no solution for Apple’s network errors outside of Apple. As many of these threads have explained. Your best bet might be to use Bonjour or another sync method. And these are detailed in the manual as well as here, as well.
Per in the previous posts (numerous times), re-read and follow the instructions in the section in the “DEVONThink Manual” about setting up Bonjour, supplemented by
where it is said clearly to set Bonjour to “accept incoming connections” on only one device on the network. No need to turn it on on your iPad.
I am making progress. thank you 
Good. As a new person to DEVONthink, one of the most important things to learn is that the outstanding “DEVONthink Manual”, the “Tips and Tricks” inside the app, and the Blog and FAQs on the DEVONtechnologies’ web site are always the first resources, before resorting to this forum, to check for answers to questions and to find new opportunities. And if using the forum, very likely the issue/problem/question already posted with the possibility of answers. Not always, but possibly. So use the search button.
Also see the “Take Control of DEVONthink” e-book, free compliments of DEVONtechnologies, on the DEVONtechnologies’ web site.
In many ways, including documentation, DEVONthink is like no other app.
Well, I am using at least 5 other apps that use iCloud syncing between MacOS and iOS devices. None of them are throwing any errors.
Just saying…
Well, I just think (paid) software should deliver on the advertised features. If iCloud Sync is on the menu, it should work without a ton of fiddling. As simple as that. 
NB: I am of course talking non-Apple apps above…
Simplistic, I’d say. DT announces that you can sync using different services. They do not provide these services. You decide to use an unreliable one – that’s not the fault of DT. You could, for example, go for Bonjour or Dropbox or WebDAV. I am, for example, using WebDAV, and I never see this kind of problem (nor any other sync trouble). [1]
It’s like buying a fridge and complaining to the company that built it when your electricity supply breaks down. Or complaining to DT if you can’t print something because you’re out of toner.
DT does not promise that CloudKit works flawlessly. It just says that if CloudKit works flawlessly, you can use it to sync your databases.
[1] Well, occasionally my Synology box decides to go to lala land and not accept network traffic any more. Not DT’s fault, though.
Whilst this comment is not directed at OP specifically, I am getting a bit fed up of people (not just on this forum) defending iCloud by saying “well it works for me”. It’s nice that it works for some people (as far as they’re aware - you don’t actually know it isn’t working until you try to find something and it’s missing….), but that doesn’t erase the varied experiences of thousands of other customers and the widely documented problems with iCloud, which Apple itself occasionally acknowledges when it fees like it. It’s not even just third party apps - there are widely reported issues with native Apple apps.
Yes, we all* want iCloud to work well. Apparently that isn’t a priority for Apple, but don’t diminish other people’s frustrations just because you’ve not experienced problems yet.
*well, most of us.
For what it’s worth, I am running the iCloud sync between Mac, iPad and iPhone, and I just ignore the error messages. I know it’s Apple being annoying, and errors usually resolve after a few minutes.
Unfortunately the error does not resolve for me. Does anyone actually know what this error-message means exactly?
I am not trusting Dropbox et al enough to use them in the longer run.
The server denied the request. Did you verify the sync store already?
i followed the instructions in the manual (pg 65/66), ie. added a CloudKit store, gave a key, told it what datebases to sync. Same on iPad.
But keep Getting “Error 15”. 
The error will not go away unless Apple fixes whatever needs fixing. Nobody here can do anything about that. You, however, could use another, more reliable sync method.
Is it a certainty that it isn’t a bug in DEVONthink’s code when other third party apps sync without issue?
As it is intermittent, a bug in the software is improbable. And the differences in DT‘s syncing to other app‘s have been been explained often here.
One of the few people apart from Apple who seems to know much about iCloud is Howard Oakley. His website has many pages dedicated to the oddities and problems it throws up. It is well worth keeping up with what he writes. Here are a couple of links:
PS: those using Sonoma might like to read this, which incidentally has an explanation of what CloudKit does: