I have seen many professor are working in a research institute and also a full-time professor at a University. How is it feasible? Are they involve in the dual job and getting a salary from a research institute and university?
I work at a government research institution in Germany. We consist of a number of institutes.
In the last decade, our president has worked on strengthening ties to close-by universities (we have institutes at different locations). As a result, there are now strategic cooperations in place. The most important one being that new institute leaders simultaneously become full professors (not "full-time professor[s]") at one of these universities. They have teaching duties but are not paid by the university (although they have a budget at the university).
Similar arrangements were in place at another research institution I've previously worked at (and to my knowledge at basically all of them, i.e., Max-Planck, Leibniz, Helmholtz, and Fraunhofer institutes).