unpopular answer... no you can't.
At the moment Finish only works with individual faces, which I agree is not really efficient.
A small work a round is to hide everything except the assembly you want to apply a finish to, start the finish feature and drag a selection over your entire assembly, apply and unhide everything again.
However, in reality you'd also not paint a complete assembly (including inner chambers, inaccessible areas, ...).
Look at the finish feature as something to exactly specify the surface area or facial individual finishes. Just selecting a complete assembly will give you inaccurate results in terms of surface area.
Also if you are just looking to just give an assembly a certain "look", you can use the appearance override.
