In general the answer is yes, however for particular cases the stresses could be the same.
Your stress "definition" (stress = load/area) is valid for uni-axial stress state, for example a rod subjected to an axial force. In this case, within the elastic range stresses in isotropic materials would be the same.
Note that this valid only if we accept some common engineering assumptions such as using engineering stresses. This means that we do not take into account how the cross-section changes under the loading due to Poisson effect. However, if we take this into account and dealing with true stresses there will be a difference due to the difference in Poisson's ratios.
Even when engineering stresses are used there are cases where the stresses will be different due to the different Poisson's ratio. For example internal forces and stresses in a plate are influenced by the Poisson's ratio.
Another example is geometrical non-linearity where the change of global geometry under loading influences the distribution of internal forces, stresses (the change in cross-section is typically not considered). Since the deformations are dependent on the Young's modulus you will get different stresses for different materials. A sagging cable is simple example for this.