My guess is it is a fastener and does not do anything useful except hold the fork together.
The right side of the fork usually contains the damper, with various adjustments. Usually, a fork has at least a rebound, but more advanced forks have one or more of separate compression and hi speed/low speed rebound adjustment (along with the internal adjustment's that can be done with spacers and oil viscosity). Often these adjusters are on the bottom of the fork leg. This fork appears to have an adjustment knob on the top. As it has no obvious branding, and cable disc brake, it is unlikely to have premium features, meaning the hex bolt is more than likely just a bolt.
You could try turning it. If it is an adjustment, it will probably turn on both directions and will feel less like a bolt and more like the knob on top. Don't remove it if it is a bolt - a good chance if you do you will spill oil, and the last thing you want is to is spill the oil from a fork you do not have service information for.
Braking shudder could be coming from the headset, worn bushings in the fork, axle or brakes. With the front brake on, push the bike forward and back and feel for movement in the headset and fork stanchions. Is the wheel correctly fitted (axle snugged up tight)? Is the brake caliper and disc correctly fitted? Have a search here how to check for these things.