Existence of the manifolds is existence in the mathematical (platonic) sense. Existence of a physical object usually means (on the level of theoretical physics) that the object can be modeled as a mathematical object having a (classical or quantum) state. A property of an object exists (on the same level) if it can be expressed in terms of expectation values of observables defined on the state space and determined by the state. For example, an electron has mass, charge, position, and spin in any direction, although the latter two have sharp values only under special circumstances.
A string in string theory has a state, and a robot arm has one (in the cotangent space of the configuration manifold) , so they are physical objects (on the level of theoretical physics). But the manifold of robot arm configurations has no state hence it cannot be said to be physically existent, though it is used to describe physical objects, namely robot arms.
Of course, existence in nature is a much more restricted matter, as in this case the model is required to match Nature suffciiently well.