Does the GS1 Company Prefix uniquely identify my company or brand?

No. The GS1 Company Prefix is not an identifier. It is a string of digits that is used as a part of GS1 identification numbers. A GS1 Company Prefix does not uniquely identify a company or brand because a given company could have more than one GS1 Company Prefix, and sometimes a company uses the same GS1 Company Prefix to identify products of several brands.