That’s not really true. A truer statement would be that the science of mechanics is the base for both mechanical and aerospace engineering (as well as civil). Mechanical and aerospace engineering, in particular, share a lot of common material, but it’s not all common. I work in a combined ME/AE department, and slowly but surely, the two curricula drift further apart every time we have a revision of our course offerings. There are a lot of similarities but still some substantial differences.
Most of these differences boil down, ultimately, to the same science applied to different in-class examples. The problem is that those examples can be so different at times, that it makes sense to have some classes taught entirely differently for mechanical and aerospace engineers (e.g. fluid mechanics). In principle, someone with a degree in mechanical engineering should be able to do any job normally meant for an aerospace engineer and vice versa provided they had a firm grasp of the fundamentals, but there will be a learning curve for some of the more specialized jobs in either field.
If he can find an aerospace company that doesn’t hire mechanical engineers, I will personally mail you a check for 1 billion boneh3ad dollars (symbol: B$). Aero-/astronautical vehicles are simply too complex to be built by a single class of engineer. Generally, any of these companies hire mechanical, aerospace, materials, computer, systems/industrial, and even civil engineers to help them solve tough technical problems. It’s all a mater of which subsystems are most interesting to a given student, which can be tough to discern for a high-schooler trying to find the best path for them. In terms of subsystems, aerospace and mechanical engineers are (nearly) interchangeable.