It has similarity in that there is a form of alliance between predominantly white fundamentalist catholics and evangelical christians and Trump which is embodied by Vance which could be seen as mimicking Putin proximity with the Orthodox church. They both use their churches to justify a civilizational agenda and frame autocracy as protection.
Still, there are several major differences one bieng the patriarch supporting Putin while the Catholic church mostly opposes Trump.
Catholics are a minority denomination in America, and an especially small minority in the relevant states.
Russian Orthodoxy, on the other hand, encompasses ~95% of Russian Christians, and there is no organized alternative to it.
... Also, Trump 2024 won Catholics by 12 points (While 2020 and 2016 was a 50/50 split.)
Whatever the church's views are, unlike the evangelicals, it's not dictating to its members how they should vote.