Yes I think it makes sense when the surface boundary layer is so thin that it can be neglected. You are assuming no drag or friction by the surface on the fluid. Or you are assuming the viscosity of the fluid can be neglected. The boundary is still there because you do not allow flow normal to the surface.
It is not required that the tangential velocity of the fluid has to be zero on the surface, that's just one possible boundary condition you can assume. Alternatively, you might choose to specify that there is relative slipping motion between the fluid and solid along the surface. Then a streamline at the surface is easy to visualize. Let the relative motion approach zero as close as you want to make it, it could be zero.
I think your original question was whether it ever 'makes sense' (or is useful) to consider a streamline on the boundary with a solid. One case would be if the contact is slipping. As a more practical example, consider a physical oceanographer wanting to model the flow in an estuary some moderate distance above the irregular topography of the channel floor. She might deploy current meters along the bottom to get an estimate of the average velocity near the bottom, and then build a model with this velocity assigned to a streamline sketched in to roughly follow the floor topography. Maybe she measures an average velocity of zero, then that could be her boundary condition.