Why is there a white line on Russian fighter jets' instrument panel?
I believe it is exactly what your instructor has said - it helps when regaining control of the aircraft during a spin.

The correct procedure for recovering from such a maneuver is to apply rudder in the opposite direction. Then, keeping the control column centered, move it forward (for an upright spin) or backward (for an inverted spin).

The problem is, when you're in a spinning aircraft everything outside is a blur and the G-forces involved can throw off the pilot's perception - this is of course especially true of inexperienced pilots. The white line painted on the control column in your picture was one method for helping the pilot to keep the stick centered. Having something simple to concentrate on helps immensely, and is more intuitive than looking at the yaw indicator.

Source: Flying at the Edge: 20 Years of Front-Line and Display Flying in the Cold War Era,
by Tony Doyle.

Edit: This book is an autobiography of a Cold War era RAF pilot, and while it does directly mention the white line painted in the cockpit, it does so in reference to Vampire and Meteor aircraft. I cannot find any reference specifically to Russian aircraft anywhere in the book, and neither does there appear to be any picture evidence of the stripe in the cockpit of any British aircraft. Perhaps someone else can source more evidence.