When using InventoryClickEvent to modify item meta, it writes it back only if moving between different inventories.

Example: Move item from player inventory to a chest, data is being written. Move item from player inventory to another slot, data is not written.

Sample code:

@EventHandler(priority = EventPriority.HIGHEST)
public void onInventoryClickEvent(InventoryClickEvent event) {
    ItemStack item = event.getCurrentItem();
    if(item == null) {
        return;
    }

    ItemMeta meta = item.getItemMeta();
    if(meta == null) return;
    PersistentDataContainer dataContainer = meta.getPersistentDataContainer();

    NamespacedKey key = NamespacedKey.fromString("testkey");
    if(key == null) return;
    dataContainer.set(key, PersistentDataType.LONG, 123L);
}

To check for the data, get the item in hand and use
/data get entity @s SelectedItem

Reproducing with attached plugin:

Get any item, try to move it inside own inventory, it will print in chat "Read: null", data was not written, as reading it returned null. If you move it into another inventory and try to move it again it will print "Read: 1", data was written, as reading it returned the data.

 
Edit: If you setCurrentItem to current item, it doesn't work too. But when you set it and cancel the event, item gets updated.