It seems like a fanciful invention from the imagination of Wes Anderson or his amazing design team. However, whilst exploring the Sächsische Shweiz we came across the inspiration for the elevator in Bad Schandau (filming for one scene was also done there).
Here it is:
For comparison, here is a publicity shot from the film The Grand Budapest Hotel :
The elevator is quite something. It was erected in 1905, is electrically powered, and was conceived and financed by hotelier Rudolf Sendig to transport guests to his Country House colony above Bad Shandau in Ostrau.
For a few euros you can travel up it, and at the top you get good views down the Elbe valley and of the small spa-town of Bad Schandau (even better when it is not a misty morning when we went up!):
At the top of the elevator (or I should say 'lift' as I am not American) they have some caged lynxs apparently, which are indigenous to the area, but extremely rare. They must have been still in bed when we were there as we didn't see them, and to be honest I am glad I didn't as it would have saddened my heart to see them caged up.
There is however a memorial plaque to Rudolf Sendig himself, the real M. Gustav of his day!