If Russians don't say Vladimir, that doesn't make his name Putin Vladimir.
The only Western nation that has a different naming scheme is Hungary, which uses the Japanese-like family-name-first scheme.
There are other countries in the West with different naming schemes than the one common in the Anglosphere, even if they happen to still have the given names first and some combination of family names thereafter; I've several times seen US outlets butcher the names of Mexican officials because they assume that the same name order applies when the order is merely superficially similar.
For instance, the headline (and URL) on this CSPAN piece:
https://www.c-span.org/video/?451988-5/united-nations-genera...
There are several acceptable ways to refer to Enrique Peña Nieto as President of Mexico, but “President Nieto” isn't one of them, and Nieto alone isn't the same kind of name as an English last name.
It has exactly to do with name ordering, since the source of the issue is that the nearest equivalent of the English last name is actually the penultimate rather than final element of the name in the order at issue.
His name uses the Spanish naming system whereby both the mother and father's name are included — like an English double-barrelled name without a hyphen — which is not affected by whether the given or family name(s) come(s) first.
Therefore, the issue is not about ordering, it's about name segmentation and culture-specific name boundaries; the issue is where given names end and family names begin, not what order they're in.
Putin's passports says Vladimir Vladimirovich Putin yet everyone is using either Vladimir Putin or just Putin, but Russians don't. (In formal letters etc).
The problem is that there are way too many differences alone in the Western world, you can't satisfy everyone's feelings.