司 shows a hand raised over a mouth: someone issuing commands. It's the mirror image of 后 (originally 'sovereign') — two officials facing opposite directions. To 司 something is to be in charge of it.
公司 'public management' is the modern word for company — the sign on every office tower. Old official titles kept 司 alive: 司机 'machine-manager' is your driver, and 司仪 runs the wedding as master of ceremonies.
A hand over a mouth giving orders — the one who MANAGES; two of them make a COMPANY.
司机 'one who manages the machine' meant locomotive engineers before cars existed — now every taxi driver inherits the title.