When an academic researcher receives a gift funding from an industry partner, does the academic researcher's university take a cut?
Than can depend on a lot of things. The answer is often yes, occasionally no. 

The question may depend, in part, on what the university contributes to you as a researcher, including office space, lab space, RA support, whatever. It might also depend on whether it is the university that has to manage the grant and verify to the funder that the funds are properly spent - accounting for the funds and seeing that any necessary taxes are paid. 

Universities have policies about these things and you should find out what those policies are in your case, but the answer is more likely that you and the university share the proceeds. The university "cut" might be quite a lot, actually.

And the characterization of it as a "gift" might be questioned or contested.