I mean, you already have a copyright to protect the game, right? the DMCA made a false claim on the word "Good Morning" on Saiko no Sutoka game on steam. you can look it up it was crazy.
Copyright is automatically applied upon the release of any perceptible tangible form of media. What you're probably talking about are trademarks. In which case, yeah, it'd probably be a good idea to register the name and logo, just in Case.
