First how cool in temperture is that closet ? Wine does not keep well in warm spaces . Get all the other stuff out of there and IMO just have wine stored to the ceiling that will give you awowo factor then add some LED lighting and a glass door .Ideally some type of cooling system too. Why not something like this from Wayfiar then you know everything will be stored properly and look very cool

Remove closet rod, its supports, and everything on the shelf.

Move the shelf down to about 6 or 10 inches above your wine rack. Give it a fresh coat of paint. It's for setting wine bottles on while you are choosing or displaying a couple of bottles temporarily.

If you have a light in the closet, hang a vineyard or travel poster above the relocated shelf.

If you do not have a light, hang a large mirror above the relocated shelf.

Add a thermometer/hygrometer in the closet to monitor temperature and humidity

I like Patricia’s idea of floor to ceiling storage with lighting and a glass door. Also the wine fridge idea is a smart one.

A closet is not the right place to store wine.

Either have a temperature-controlled space for it, or admit you want to show off the wine you have. In that case, several wine display racks can be considered and they can be mounted directly to the wall.

I have a few questions...

What are the dimensions on the interior of the closet? width x depth x height

Are you willing to match the height of the adjoining door?

How many wine bottles will be stored?

Do you want to store anything other wine related items in the closet?