Are postdoctoral scholars considered students or researchers in other countries?

In general, if you are not enrolled in any course and also are not enrolled in a program in which you may eventually earn a degree, you are not a "student."

I think postdocs are assumed to be employees as research staff (not students) of each university , as far as they have contracts with universities and may receive salaries. They do not take courses in the university. They conduct some researches in a research group in collaboration with professor(s) there. Their research topic may be something very close to their PhD project, something near to their PhD research in other concept (such as interdisciplinary projects), or something very different.

PS: Some PhD students also have research contracts with universities and are assumed to be employees of those universities, but I think their difference with a postdoc is that they receive a degree but postdocs are not receiving any degrees.

It's also relatively common in the USA for post-docs to have non-university appointments. There are many government and private labs that hire post-docs to do research as employees. These positions have various levels within their respective companies/labs.

For instance, I am a postdoc at a USA national laboratory. Here, postdocs are considered long-term temporary employees (with some reduced benefits and a reduced salary compared to a full staff member as you would expect). We aren't considered students or under any particular education program, though there are development programs offered to help in the transition between a full-time student position and a full-time research position.