From afar it seems to be another example of government and religion symbiosis, like what Pakistan did in the 60s/70s.
A country starts out with a certain amount of religion built into its culture.
Government wanting to retain power plays up its chumminess with certain groups, typically fairly intolerant/doctrinal ones, of that religion.
"You support us (the government) in the name of culture. We'll support you by allowing you to repress what you don't like and giving you more say in state affairs".
Given enough time and propaganda the coopted religious group becomes more important, possibly even in its position vs different subgroups of the same religion.
There is no real way to "prove" this thesis, but this trick has been played before, from Franco in Spain to Pakistan to Saudi Arabia. Yes, and Poland too nowadays.
South and Central American dictatorships struggled with using this against peasant and Communist movements in the 70-80s because South America's Catholicism was influenced by Liberation Theology, but otherwise it's a frequent item in the dictator's bag of tricks, along with playing up external enemies.
And, yes, I'll go along with the idea that the USSR was only nominally atheist because it was a foreign imposition, not a natural cultural fit to Russia. But Russia's "normal natural religiousness" still doesn't seem to explain the evolution towards what seems to be a fairly intolerant version of Christianity, especially after a long latency under Communism.
Somewhat related is the idea that Eastern Europeans tend to have more conservative religious views than Western Europeans (yes, this link concerns Catholics only, but the point is to compare geography-based conservatism which is hard to if you compare varying flavors of Protestants and Evangelicals or if you look at the Orthodox Church which is right now influenced a lot by Russian prominence):
Catholics in Central and Eastern Europe tend to be more socially conservative than those in the West. In every Western European country with enough Catholics in the survey for analysis, majorities of Catholics support legal gay marriage, including overwhelming shares in the Netherlands (92%) and Belgium (83%). By contrast, most Catholics in Central and Eastern Europe oppose gay marriage, including nine-in-ten in Ukraine and Bosnia. Catholics in Central and Eastern Europe are more likely to favor legal abortion than same-sex marriage, but support for legal abortion is still substantially lower than among Catholics in Western Europe (median of 47% vs. 71%).
There is risk in this type of alignment, as a church can lose a lot of moral stature if the government falls later. The Spanish Church still get tarred by Franco's support, even if it's hard to separate from pushback against child abuse. By contrast, the Polish Catholic Church has done rather well from its role in opposing Communism throughout the 80s.
Last, "being a Christian country" can mean different thing depending on who you ask. When it comes from a government engaging in this type of behavior, who is it speaking for? The government and their committed religious supporters? The masses of variously practicing religious people, from the devout to people merely born into it? The people who are not religious or who follow other religions?
In the case of Russia, in 2012, 47% were nominally Christians.
In the absence of clear polls or classifications, it is easy to make these claims from a government pulpit. Some factions of US politicians make this type of statement frequently and have to tried upon it, despite a strong constitutional separation of Church and state. Does it mean everyone there believes in God, the Bible and the Devil?
I.e. the party in power can make these claims and gradually influence public belief without the need for initial mass conversions. In a sufficiently illiberal political system they can move the needle quite a bit over time, partially by just ignoring and suppressing anyone who does not follow the professed state religion.
If no one has the right to broadcast "no, we are not a religious state" and if there is little judiciary oversight, then the state can become religious in practice at a state level, regardless of the doctrinal beliefs of individuals.