Is it necessary to have different keywords on different pages per website? Is that even allowed and an acceptable SEO strategy?
Yes, it is necessary to target different keywords on each page of the website in order to avoid clashes between web pages. If you will target same keyword in two different page than Google gets confused as to which page to rank for particular keyword.
Depending on the keyword density and how you have promoted two common page of the website. Google decides priority between two pages.