This is a shot in the dark, but I think you are...
An arrow. (Yes, "shot in the dark" is a pun).
Melody admirer I am, not many sense the truth,
An arrow is launched from a bow. In a way, it is a "stringed instrument," although most people would not consider it so.
I belong to everyone, old and the youth.
Arrows were used by ancient civilizations, and are still used today for hunting and sport.
While that little guy stands upright,
Usually, the archer holding the bow is standing.
I jump atop, while gravity joins the flight.
Arrows are launched in upward trajectories, and gravity makes them arc downward.
I own a skull, but struggle to think,
Some arrowheads are made of bone. A "head" made of bone is a skull.
Have an amazing eye, but struggle to wink.
This is a creative one. It uses "have" in the sense that the arrow "has" an archer who launches it. The archer needs good aim, and winking means they can't see as well.
I do have a talon, and yes it plucks,
Another creative use of "have". The archer fires the bow with their finger ("talon") to pull and release the bowstring. This is like plucking a string.
Having a neck with no food sucks.
The shaft of an arrow is like a neck, because it connects to the arrow "head".
Dare you try provoking my head,
The arrowhead is typically sharp, and is what causes injury.
Every now and then I fancy the bloodshed.
Arrows were often used in wars, for ranged combat.
I ain't precious though being owned by a lord,
Individual arrows are cheap. However, a lord's army of archers was incredibly powerful, especially in ancient warfare.
Modest I am, anyone can afford.
Again, individual arrows are cheap. You can make an arrow from a thin strip of wood, plus a rock or sharp bone.
I've spent ages with the white stones to thrive,
Multiple possible meanings. Bones and rocks may be used as arrowheads. Or, the dead who are killed by arrows leave behind bones.
Signs of love were embossed then and for eternity they will survive.
Arrowheads crafted from stone required lots of time and focus for carving.
Listen carefully, I need to warn you,
If you have very keen hearing, maybe you can hear an arrow in motion.
I arrive with M plus V-squared by two.
I'm interpreting this as (1/2)mv^2, the formula for kinetic energy. The reason an arrow projectile is so deadly is that it has kinetic energy carrying it forward.