After a bit of trial-and-error, I discovered that the Update URL provided in the Knowledge Base for FritzBox Custom DynDNS update script is wrong.

The correct address - as per API documentation - should be:

http://api.dynu.com/nic/update?hostname=[HOSTNAME]&myip=[IP ADDRESS]&myipv6=[IPv6 ADDRESS]&password=[PASSWORD]

Confirmed working on FritxBox 7590 with FRITZ!OS 07.27

Great! I have been trying to get this to work for a while now, only resulting in bad auth.
Your solution works!

Someone should update the help page...

Thanks again!

Thank you all for the information. The knowledge base has been updated. We rely on customer feedback to improve our service. Give us a shout any time!

Is this the same Update URL for a Group Login with Fritzbox?