Toyota representatives confirmed to The Autopian that this is the same Mazda branded colour, thanks to the joint plant of the two Japanese automakers in Alabama (USA). The paint composition is identical to the original.
The Soul Red Crystal colour will be available even for the base Corolla Cross in the US - it's not exclusive to special editions. By the way, the same shade is used for the Mazda CX-50, which is also assembled at the Mazda Toyota Manufacturing plant.
Soul Red Crystal has a complex structure of three layers, and it has been used in this form since 2016. Here's an old Mazda video dedicated to this paint:
However, the cooperation between Mazda and Toyota goes far beyond painting: the companies are actively exchanging technologies. For example, the Toyota Yaris is produced as the Mazda2, and the Mazda CX-50 hybrid system is taken from the Toyota RAV4.
Source: The Autopian