Yes.

FF cameras began decades ago, with the introduction of 35mm film for still shots. The vast majority of film cameras out there use 35mm "sensors" (film).

With digital, we just replace the film with a digital sensor of the same dimensions, at the same point in the camera body.

Issues with light fall-off in the corners can be a concern if the maker doesn't take it into account in the microlenses (on the sensor), but Nikon and Canon both handle this just fine.

Cheers