I've seen lot of videogame title screens show a copyright notice together with the company name. I'm about to publish my game and I am looking for possibilities or requirements on how or when to add such a copyright notice (including date or year published). Does it apply to both indie game makers and larger companies?
There is no requirement. Copyright is automatic the moment you produce a creative work. An explicit copyright notice is optional and not required to claim a copyright later. It is just a friendly reminder of who the copyright owner is and that they care about the copyright of their work.
When you want one, you can place it anywhere you like.
When someone violates your copyright, a copyright message can serve as a proof that you are the original author. So you should try to integrate it into your game in a way that removing it requires effort.