What is not clear for me, is it mandatory to have the resulting product also under a permissive open source license?
Generally, no. You are required to distribute the licence text, but there is no requirement comparable to eg GPLv3's s5c, "You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy". However, you may not free your downstream recipients from the permissive licence's obligations, and the easiest way to do this is to redistribute under the same licence, or a less-permissive one.
If it is not mandatory, how it differs from "do whatever you want", I mean except, the expectation to show the original license when distributing?
You must also generally preserve copyright notices, and Apache2 has some additional text about not enforcing your patents against licensees. But apart from that, you pretty much can do what you want: hence permissive licences.
Say the company do not distribute it at all, just uses it in modified form as a part internal commercial production process, say banking industry?
Most permissive licences don't distinguish between your obligations on modification, and your obligations on redistribution, so not distributing makes little difference.