In an interview on RT Putin said, "I really doubt that Stalin, in the Spring of 1945, if he had a nuclear bomb, I doubt that he would have dropped it on Germany. In 1941 or 1942, when it was a question of life or destruction of Soviet state, maybe he would have, if he had it. But in 1945, when it was clear that the enemy was in capitulation, essentially, when the enemy had no chance of winning, I really doubt he would have. But the Americans did this against a defeated Japan, a non-nuclear country". Could Stalin, who was responsible for mass repressions, ethnic cleansing and the execution of hundreds of thousands have been able to act sensibly when it came to mass annihilation of an enemy by a nuclear bomb?
Your scepticism is well-founded. Putin is engaging in propaganda, trying to make Stalin's USSR seem like a state with moral superiority over Truman's USA. We can't know what Stalin would have done, but some things can be deduced.
If he had a lot of atomic bombs, he might have used some on Germany. If he had only a few, he would have wanted to keep them as a hedge against the aggression he feared from the Western powers, particularly the USA. It was clear by early 1945 that relations between the West and the USSR were going to be difficult, and Stalin was aware of the Manhattan Project via espionage.
Given a lot of bombs, what would he use them for? Probably not busting cities, because they contained factories and other industrial plant. The Soviets dismantled large quantities of that in their occupation zone and took it back to the USSR, as self-awarded compensation for what the Germans had destroyed. Destroying it with nukes would have been counterproductive.
Also, Stalin wanted to capture Berlin, to back up his narrative that the Soviets had defeated Germany themselves. The symbolism of the capture of the landmarks of central Berlin was important for that, and he was more willing to sacrifice his soldiers for symbols than the West were. Eisenhower was willing to let the Soviets have Berlin, not considering it worth the lives it would cost for him to get there first.
So Stalin's use of nukes would have been limited to German troop concentrations outside cities, essentially tactical uses. The large and heavy atomic bombs of the late WWII era were hard to use for such work, because they needed large bombers to carry them. Putin seems to be trying to create a moral superiority from a tactical limitation.