On paper, the Russian government's political system looks quite similar to that of the United States, although, obviously, it isn't identical. Both have an elected President and separately elected legislators, and there is some degree of checks in balances embedded in the Russian constitution, although it isn't quite a fragmented as the U.S. Constitution. Neither constitution vests unilateral power to wage war in the President himself, but both do give the President the highest level of executive branch authority over the military.
Some of the devil is in the details. Russia is a hybrid between a strong Presidential system and a Westminster style parliamentary system, while the U.S. puts more distance between the cabinet and the legislature than Russia does. But the differences that do exist aren't really very material for the purposes of this question.
The answer is more a matter of the real world reality in which Vladimir Putin, whose current title is President of Russia, has de facto control of all material sources of political power in the country at the national level. This has great similarities to the manner in which the formalities of the political system in the Soviet Union were irrelevant because the Communist Party controlled everything from behind the scenes.
While Vladimir Putin most recently ran for President as an independent, his primary base of support and historical partisan allegiance in Russia is the United Russia Party which has controlled the national government in Russia since 2003. His close ally and the current formal leader of that political party, Dmitry Medvedev served as President of Russia from 2008–2012 (after which Putin served as President, he is currently in his fourth term) and as its Prime Minister from then until 2020, when a new United Russia Party Prime Minister began to hold that office. Vladimir Putin won 76.7% in the most recent Presidential election with United Russia's endorsement, United Russia holds 324 of 450 seats (more than two-thirds) in Parliament, and 60 of 85 Governors (almost three-quarters) in Russia (a position roughly equivalent in the organizational chart of Russia to a U.S. state governor) are United Russia party members.
Some of this control has been secured by extralegal means and political tactics that are dubious at best and probably involve illegal misuse of government resources and non-governmental criminal activity. These include side agreements with Chechen rebel leaders to have these rebels act as an unofficial militia for Putin, deals reached with "oligarchs" that trade favorable government dealings for loyalty, unfair rulings controlled from above in election administration, abusive uses of the Russian criminal justice system to persecute opponents, and harms experienced by political opponents with no official governmental involvement up to and including death and threats of death to the politicians and their families. Such tactics are hardly unexpected from a man who built his career prior to his involvement in politics (which started in the year 1990) in the notoriously ruthless KGB of the Soviet Union where he served from 1975-1999 first as a counter-intelligence agent rooting out spies in Russia, and then as a spy and covert operative in East Germany.
A complete catalog of how Putin came to be the dominant figure in Russian politics and specific instances of the tactics that he or others in his political movement used or are widely believed to have used would probably be too long to fit in a single Politics.SE answer of its own limited to that topic and is beyond the scope of this question.
For purposes of this question, how Putin's pervasive political authority came to be is just window dressing, and the material point is that he does indeed have such political authority and practical political power in Russia. His dominant power in the Russian political system is a point that even someone who takes umbrage at allegations that he acquired that power by improper means would generally not dispute.
In practice, the reality is basically a dominant party system. Putin's party and members of his coalition aren't the only legal party and dissent is not entirely suppressed as it would be in a totalitarian system of government. But, the control that he has, directly and through close and loyal allies, is sufficient, that he has near completely and absolute political power to do anything within the capacity of the Russian government.
Vladimir Putin certainly has, as a matter of practical reality, the ability to unilaterally decide to start a nuclear war. So does U.S. President Biden. This is true even though neither man has the formal power to declare and wage a war without legislative involvement.