Here are some thoughts on how souls work in MY version of Narutoverse.
In this version the core of a soul is consciousness and a Soul can only have more than one core for so long before splitting apart and becoming independent souls.
Also multiple souls can't coexist for long within the same body unless all but one are sealed.
This also explains what happened to inner Sakura.
During the first chuunin exam her second consciousness aka schizophrenia helped her push Ino's mind out of her body. This is only possible if it is an independent part of her mind aka an independent consciousness.
At this point in time it was still in the process of splitting apart from the main soul so it could still stay within her body.
Later it either merged back with her main soul and disappeared because she overcame the mental trauma that caused her schizophrenia in the first place or it fully separated from her. At this point it would have to leave her body as two souls can't coexist in the same body for long.
That is why we never see inner Sakura anymore in Shippuden.
By the same logic Hyorinmaru being a separate consciousness independent of Toshiro would slowly form his own soul around the secondary core.
This process takes quite a few years as it has to split off a part of a considerably strong soul that can sustain a dual core for about a decade at most.
After that it would be a battle between the newly born soul of Hyorinmaru and Toshiro over possession of the body with the loser being expelled from the body and thus dying.
In order to prevent this scenario Mito used fuinjutsu to speed up the process and then seal the newly created soul within his body like a bijuu. That way the two souls can coexist peacefully within the same body.