I was curious myself so I looked this up on wikipedia.
The USSR’s “coast guard” was the Maritime Units of the KGB Border Troops.
After WWII, the Soviet Border Troops basically served the function of both the US Border Patrol and the US Coast Guard, with different units for each. The Border Troops were put under the NKVD and became the NKVD Border Security. Later they were put under the KGB and were renamed the KGB Border Troops, where they remained until the fall of the Soviet Union in 1991.
Currently, the Border Service of Russia (including the maritime units which serve as Russia’s “coast guard”) is under the Federal Security Service of Russia.
Sounds like they have the FSB Border Troops Coast Guard, and they also have the National Guard Naval Service Corps., who it looks like may pick up some of the search-and-rescue type duties that the US Coast Guard handles.