Is there any disadvantage with transmitting the same excitation signal to several resolvers (basically rotating transformers)?
Is there any disadvantage with transmitting the same excitation signal
  to several resolvers (basically rotating transformers)?


Well, going back to some old-fashioned positional systems used on-board ships (for instance), that's exactly what they did and there was no option. One synchronous resolver was operated manually from some desk /back-office situation and it remotely turned (as an example) an antenna fastened to its interconnected resolver. This type of system has no-option other than to share the same excitation signal so, the precedent is already set: -

 

Image from Moog synchro handbook page 2-2.

The last point mentioned in the article above (multiple receivers decrease accuracy) is not about R1 and R2 sharing but about S1-S2-S3 sharing and this does not apply when doing an "electronic" decode of position. This question is about R1 and R2 sharing.

So, the precedent is set but the question then sort of boils down to how many different loads can you connect to a power supply without creating some disadvantage. As EEs we should all know about these things: -


Don't overload a power supply (AC or DC)
Don't use high inrush loads that might upset other loads using the same power feed.
Don't generate too much noise that can back-feed to other loads.


All these problems are solved by good design practices.