The prince still goes by his given first name. However, to avoid criticism he chose adopted a new last name (Rivers) and avoids mentioning his past. So not too secretive but enough so... if that makes sense.

As for why the assassin and prince would work together, that does prove to be difficult. The assassin's reasoning may be that it would be easier to find the thief if he teams up with someone and then just later kill them -- if that's the type of man he is. However the prince wouldn't have the same reasoning. Unless he suspects the assassin for having connections to the palace. And the prince would rather team up and keep an eye on the assassin. Or some other idea if you can think of something. :)