How could duplicate entry on unique key happen if only one master is active?
Do not see the problem. When somebody enters the email address a second time then this happens. Nothing to do with replication. This is what the unique key is used for.

If this error would break the replication then I do not understand how replication could ever work. Every user could easily break the replication by, in this case, entering an email address 2 times.

I am not an MySQL expert but I think that the doc relates to errors other then maintaining data integrity. Perhaps somebody with more in-depth knowledge of clustering can help you on how to configure master-master clustering.