skype wants to export "Apple ID Authentication" from the keychain
I seem to have fixed this... unfortunately I don't know what part of what I tried was the actual key to the fix.

I spoke to Skype support, live chat. They hadn't heard of the issue - though tbh, 1st line support are always going to be 'last to know'.
The best I could get was 'Have you tried reinstalling it?" I said, it's a Mac, reinstalling isn't going to make any difference.
We tried it - it didn't. No surprise there.
I did it again, this time using CleanMyMac to get all the prefs &amp; hidden stuff, kexts etc; rebooted &amp; tried again. It still remembered my login details so I'm not sure how successful that was.

I looked in keychain to see what was in there. I was surprised at how many bits &amp; pieces there were.
Having made sure I knew my login details, I deleted the whole lot; everything that came up when I searched 'skype' - maybe 10 or 12 keys - &amp; rebooted again.
Skype had lost my auto-login... good.
I could then see which new keys it added as I completed a manual login.
There were only two keys - Skype for Desktop &amp; Skype for Desktop MSA. After several days there are still only two.
No functionality seems to have been lost. Conversation histories etc appear to be stored at Skype, not on the local Mac.

It hasn't asked me to export anything since.

It would probably be useful if someone else with this issue could try deleting only the Keychain entries, or only doing the 'clean' reinstall; reboot &amp; see how that goes. See if we can figure whether it's the 'clean' reinstall or the keychain deletion that actually fixes it.

Note: This machine has a complete line-of-sight migration path back to my first OS X Mac in 2005 &amp; some data from before that, when I migrated from OS 9. Skype has been on it since it was invented, so I may have a whole lot more historic cruft than most people.