Exemption for Dependents of Employment Income Earner (for Part-timers)

I am a student, I am on a student visa (留学).
I just started part-time at two companies (A and B), and Company A asked me where I want to submit the application for the Exemption for Dependents of Employment Income Earner (扶養控除等). They mentioned that I should submit it to the company that I am earning more from. In this case, I will probably submit it to Company B. However, Company B never mentioned anything about this.

I am really confused about this whole thing because I’ve worked some part-time jobs at some apparel shops before and nowhere asked me to do this (I got hired properly).

If anyone knows how this works, please let me know!