Would their soul become permanently a part of the Original Caster's body? - No
Even if the souls of the simulacrum and the caster were the same, the soul switching is still due to a spell effect from magic jar. The fact that the souls might be the same (as might be more compellingly argued with the clone spell) has no bearing how magic jar operates.
Thus, the normal effects of magic jar apply:
If the container is destroyed or the spell ends, your soul immediately returns to your body. If your body is more than 100 feet away from you or if your body is dead when you attempt to return to it, you die. If another creature's soul is in the container when it is destroyed, the creature's soul returns to its body if the body is alive and within 100 feet. Otherwise, that creature dies.
Addendum #1 - A simulacrum may not even have a soul
See Does a simulacrum have a soul? for more discussion on this.
Addendum #2 - The soul of the simulacrum and the soul of the caster are probably not identical
Assuming the soul (a not defined entity in 5e) consists at least partly of a creature's consciousness, memories, and personality, there is a good argument to make that the souls are definitely not identical. For one thing, nowhere in simulacrum does it say that it has any of knowledge or memories or personality of the creature it duplicated. Another spell, clone, does do this so it is not unreasonable to think that this omission was intentional. See What does a Simulacrum know? for further discussion on this.
Addendum #3 - A simulacrum probably would not try to take their controller's body without being instructed to
The simulacrum is friendly to you and creatures you designate.
Generally, things that are friendly to one, don't try to steal one's body against one's will. I would say that, in most cases, this would be an openly hostile act against its creator.
It obeys your spoken commands, moving and acting in accordance with your wishes and acting on your turn in combat.
I think it is generally assumed that having one's body stolen is generally "against one's wishes".
It is hard to think of a case in which a simulacrum could reasonably pull this off without violating the intent if not the explicit reading of the spell's effect.