How exactly do you assign a specific item? I only know how to give a uniform, and to remove a specific item.
You may need additional things like a bookkeeper who has accurately counted all your items, but if you have met the requirements -
press m.
down arrow until the name of the squad you wish to equip turns blue (this assumes you've made a squad with dwarves in it)
right arrow, then down arrow until the name of the dwarf you want to equip turns blue
press e
to equip shields, press S, for weapons, W (capitalization matters)
on the right side of the screen you now see a list of choices. The top one should be specific shield/weapon, already highlit with blue. If it is not on the list - something is wrong, perhaps bookkeeping. Press enter while specific shield is blue.
You will now see the complete list of every shield/weapon in your fortress that is not currently marked as forbidden or for dumping. This does include all the shields/weapons that are currently being used by other soldiers, so you will need to keep track. The game will not give you much information (that I've been able to find) to help you understand your error if you've inadvertently assigned the same item to 20 dwarves or something.
Use up/down arrow to slowly scroll the list or pageup/down keys to quickly flip through it. Highlight the desired item in blue and press enter.
Note this can be used to specifically and individually equip each bit of each dwarf's equipment, if you wanted to. Check the top of this screen (the m-e screen) to see all your equipping choices.
P is especially nice - it lets you see each specific item, including wear, decoration, and quality levels that the dwarf currently is 'assigned'
Hope this helps!