Well, yes, Flag of Convenience is a thing.

But there is a "but", which is that in the articles of war, the flag of a ship does have quite a few implications. E.g. when two nations are at war, stopping ships flagged as belonging to the opposition gives certain rights of stopping and searching them, blockading their passage, seizing the vessels and cargo, etc.

And the relevant characteristic in that case is the flag, not the captain's nationality: > Art. 51. Enemy character. The enemy or neutral character of a vessel is determined by the flag which it is entitled to fly.

http://hrlibrary.umn.edu/instree/1913a.htm