Copyright is a legal protection against copying and only registration makes it real.
Actually in most countries copyright occurs automatically at the point of 'permanence' or 'fixation", which to put it simply, is as soon as its more than an idea. Even just a GDD would qualify for this.
Most countries do not even maintain an official register of copyrighted works, generally only trademarks and patents are registered. The US does have a federal copyright register, however it is voluntary and isn't a requirement for a work to be 'copyrighted', as the US is also a participant in the Berne Convention which is where international copyright law originates from.
Here in the UK we have no official copyright register, but there are private services that offer '
copyright registration'. In reality all they're doing is publishing that so and so, created x and the date that it was '
registered'. These aren't official in any real capacity and are only really of use as evidence if you end up going to court. Are these services worth it? - I dont think so, anything date stamped online (such as a facebook page or post) has essentially similar weight in court. Plus the mountains of supporting work and documents most games have would be more than enough to prove ownership once in court.
There is also no legal requirement to feature a copyright notice on a work for it to be considered copyrighted. So even if your work has no legal notices on it what so ever, it is still your copyright.
*Disclaimer* As a designer by trade who does a lot of freelance work I've made it my business for a long time to be very aware of IP law. However I am no IP lawyer. If you've got a valuable IP read up on your country's specifics in depth and if necessary consult with a lawyer that specialises in IP law.
For other brits out there, here is a simple overview of our IP laws in the UK:
https://www.gov.uk/copyright