You don’t need CPT’s to display custom information. You could use custom fields (you may need to enable them in the screen options at the top of the post edit window). You’d just need to include the custom fields in your loop but position them so they appear over the sidebar. Their information would be related to the post their on and you wouldn’t need to create a new sidebar for each post.
The same thing could be achieved with custom meta boxes. I’ve heard good reviews on the plugin below but haven’t used it myself.
http://wordpress.org/extend/plugins/verve-meta-boxes/
Adding to what you have mentioned below it is possible to add a description to a tag so you could just format all of the extra information in the description field and use get_the_tags to call the information you want. Then you’d just have to add the tag you wanted to the post.
Taking it a step further you’re probably better off registering a new taxonomy (Game) setting it up so it works like a tag and calling the description of that. Then you won’t be cluttering up your normal post_tags with the game names and it will be easier to call the description on only that taxonomy instead of all tags (and trying to exclude the ones you don’t want). This would also let you use the new taxonomy in other spots in the site and thus relate multiple types of content (if you ever decided to add a new content type).