Sounds like BS.
Ask for it in writing, quoting which part of the SA Firearms Act they are referring to. I searched through the current act and couldn't find anything.
There is reference in some states firearms application process that you need to show you are a regular resident of the state, to acquire a license in that state. Perhaps that is what they were attempting to say?
As for shooting in multiple states, I believe all states except WA recognize interstate licenses. This is how you would travel/shoot in multiple states. Your license entitles you to possess a firearm in those other states, you just need to follow their particular transport and storage laws, of course.
As an example, from the SA Act.
Part 3—Acquisition, supply and transfer of possession of firearms
22—Trafficking in firearms
(1) Subject to this section, a person who acquires a firearm is guilty of an offence unless—
(a) the person is authorised to acquire the firearm by a permit under this Part (or under corresponding legislation of another State or Territory of the
Commonwealth); and
(b) there is compliance with the prescribed process for acquisition of the firearm.
Write them back and ask what Part and Division of the Act they are referring to. I bet you get nothing.