Regarding specifically the USSS's role as security and protection personnel for the US president, vice-president, presidential candidates and foreign dignataries.
on the second part of the question, "the black-clad Secret Service agent taking a bullet for the president" and "get down, Mr. President!" are major tropes in the Western world. If the USSR did have such an analogue, did it play a similar cultural role? If it didn't, what other agency "filled the gap" for it, so to say?
The comparable agency in the Soviet era was the Ninth Directorate of the KGB, which provided personal protection details for senior Soviet officials and guard forces for critical infrastructure and government installations. After the fall of the Soviet Union, the Ninth was broken out into a standalone agency, not dissimilar to the Secret Service, now called the Federal Protective Service.
The Ninth Directorate had nothing like the reputation or presence in popular imagination of the Secret Service. The KGB is most identified with the work of the First Chief Directorate (foreign intelligence) and the Second Chief Directorate (political policing) — though it was in reality a vast bureaucracy responsible for everything from SIGINT (the Eighth Chief Directorate), to censorship (the Fifth Chief Directorate) and border security (the Border Guards Directorate.)