There is a big difference between The Catholic Church and Orthodox Churches like the Russian one. The Catholic Church claims dominion over the rulers of countries while Ortodox Churches are classically under the local ruler. So the partnership Putin is going for would be much more uneven and one could already see how the Russian Orthodox Church is already an instrument of power for him and a way to advocate for limiting people's liberties which is a big thing he's doing to broaden and cement his own power.