There is no single person responsible for Cyrillic script. It is mostly believed to be created by mixing and changing Greek and Glagolic scripts by the scholars of Preslav Literary School, which was indeed in Bulgaria. After a while, Peter the Great changed it a lot. And then Stalin stomped out almost all the deviations in the usage of the script.
The last part is mostly why it is considered Russian. A lot of languages suffered because of Moscow just forcing them to use the version of Cyrillic that Russians were using.