Question

Off line testing of a site should be done on a single browser only.

Answer 1

No, the statement is incorrect.

Offline testing of a site should be done on as many browsers as possible.

Not only multiple browsers, but also multiple devices.

Nowadays, it has become a need of the moment to make the sites responsive (i.e. accessible for as many devices as possible.) since a major sector of population uses internet via devices other than personal Computers and laptops.

So, it is important to check how the data is arranged  on the screen in different devices and if the cached elements are working properly or not.