I was the engineering manager for a time at a company who launched a competing product to dropbox (its dead now). During my time there we deployed a common c++ core that did the business logic, and we wrote the uis in Qt with Qml for the forms. We also used swig to export the c++ objects to java. Our ui forms looked native, and they ran on every mobile os at the time (Windows phone, iphone, android), and we even used some of the Qml forms on our desktop ui.

Yes it was kind of a pain in the ass to share code like this, but it worked, and the mobile guys mostly just dealt with the specific problems relative to the mobile domain and not api integration issues with our cloud.

That being said, I can tell you for certain our mobile devs (most of them anyway) would've preferred writing their own client api in the native language of the platform. Maybe Dropbox just got tired of dealing with that, they have the money now to hire fleets of engineers to write just about anything on any platform, why would you even bother sharing a client library in that case? But in our case, we were a small team, so we really didn't feel comfortable just letting the next mobile dev who walked in the door come up with another foundation library.

Coding is hard, coding with a team is even harder.