Is a basic network cable tester enough, or is certification testing something different?

They answer different questions.

A basic tester is very useful for finding opens, shorts, crossed conductors and other wiring faults; a cable certifier goes much further and measures whether the installed link actually meets the performance limits for its category.

If somebody wants evidence that the cabling is genuinely Cat5e or Cat6 rather than merely wired in the right order, certification is the relevant test.