There has been a bit of hubub recently over the licensing of the Common Core Standards, and how it might be used to enforce "alignment." Here's my quick "I'm not a lawyer" analysis.
According to their "public license," which grants "a limited, non-exclusive, royalty-free license to copy, publish, distribute, and display the Common Core State Standards for purposes that support the Common Core State Standards Initiative." So this means that theoretically there is some range of uses with is neither fair use, nor allowed by the license. In practice, this is a thin, unimportant slice, since most uses are likely to be educational and would not affect the "market" for the standards themselves.
The real question is whether the NGA and CCSSO hold any trademarks associated with "Common Core." What would stop someone from calling their product "Common Core Aligned" is trademark law. I can't stamp "Microsoft Certified" on whatever I want not because of Microsoft's copyrights, but because of its trademarks. Nothing on the Common Core Standards site indicates that they hold any trademarks, while the other Common Core (the curriculum) clearly does assert a trademark on the phrase. Nor does anything show up from NGA and CCSSO show up on a naive trademark search.
Presumably they could pursue a certification mark at some point. That would at least give them control over a specific "Common Core Aligned" or maybe "CCSSI Aligned" logo.