Front end motorcycle wobbly, feels "loose"

I would check the following (with no weight on the front)

1. check for play in the wheel bearings - the wheel should not rock side to side - Are there spacers required when installing the front wheel, could one of these have been missed/replaced backward? I had that happen on my bike before (once backward and another time lost entirely)
2. check for play in the front suspension - again these should not allow lateral/front to back movement, the only real movement should be along the axis in terms of damping/recoil no other movement should be felt (minor flexing is allowed).
3. check for play in the steering head bearings - there should be no rocking/clicking, nor should they be so free the handlebars flop around.

In addition to the answer posted by @Mauro, with which I agree fully, I'll add that motorcycle fault diagnosis is often challenging because the rider "feels" something in one end of the bike that's actually caused by something at the other end of the bike.

Thus, you should also check the rear suspension. Are the fasteners all present, and tightened to the correct torque? Is the rear shock in good shape with adequate damping (both compression and rebound)? Are the shock's mounts in good shape and torqued correctly? Are the swingarm mount pivots in good condition and adjusted to the correct amount of clearance? Is the tire in good shape, correctly inflated, and displaying no signs of failure or delamination? Is the rear wheel true, and showing run-out within specification? Are the rear wheel bearings smooth, and and free of play (i.e., demonstrating no lateral tire movement, as Mauro describes in his answer)?