Thanks for your quick reply Asrujit!
I did have a look in the server_config table and there is an expiration date in there, so the gdb would have to be re-authorized at some stage anyway. I'm still a bit confused that the upgrade worked because this basically means that once a geodatabase has been authorized, it doesn't care about what version it is - it'll just continue to work for as long as the initial authorization key (which may be of a much older version) is valid. I would have expected there to be some kind of dependency on the software version as well.
Based on your reply I will install the ArcSDE command line tools and re-authorize all geodatabases for version 10.2 after the migration, just to keep things tidy.
Regards,
Jan