So only recently found this and while I still haven't processed the last episode (or rather the description of it), I think I have a theory on what happened here. To me it seems like the show's creator had an original IP involving the AH and used the Fandorin property for the sake of generating 'interest' etc.
While at least one of the brothers read the book especially considering the wedding attack etc considering how the show mostly focuses on its original plot & characters etc to me it seems like brothers just piggy backed off of Fandorin.
I think that explains all the 'AH' details they did get 'right' since that might have been the main focus of the creators. Now on the show's politics while it may not be a direct result of censorship etc I think considering the positions the creators seem to clearly want people to identify with, especially with who the villain is, I will say it does seem to be a product of Putin's Russian.
We are meant to be against the plan for fully free elections and sympathies with the all powerful Tzar and the interventionist uncle is meant to be right.