I have searched the web for examples of such cake, without success. Has anyone ever made a cake like this? Any ideas how to approach it?: cover each half separately, use an airbrush?

There'd be a number of different ways.

One way is to use buttercream and first pipe the join where the two colors meet first and get it smooothy smooth then continue piping to cover the cakes or apply the rest with a spatula and then smooth. Because you get the best chance to keep that two color edge clean and even. I'd use an open coupler to pipe it.

Or apply fondant on each half overlapping the join slightly and then take scissors and cut through both colors and then remove the excess and snoothy smooth ithe seam down pretty.

I'm assuming you want the seam to be all smooth and seamless in appearance. If not just apply whatever icing and pipe over the seam or apply pearls there or whatever you'd like to cover it.

You were asked to make a black and blue cake??

Maybe you need to explain the significance of this to the couple getting married? That they should avoid this theme?

Well, I did not explain myself enough - this is not a wedding cake, but a his and hers birthday cake when their respective birthdays are just one day apart!

Sorry--I was using a library PC and was getting close to the early closing time.  I have now reconfigured my new PC at home and can read at leisure.

I would advise the customers to stick to pale blue if possible, or else a cake covered in white with blue and black trims.  That comes across as being more appetizing.  Not to mention, easier for you to put together.

If you have to cover in two colours, then you cover about 3/4 of both sides with a complete overlap.  Then  you take a craft knife and gently cut through both layers so that the seams butt perfectly.

Or you could airbrush, using a mask to prevent spatters.

Thanks all for the ideas and suggestions. I will do my best to convince the client to work on a different idea. 

I'd use two colors of fondant, because that much food coloring of black and blue is going to be nasty to eat.