There are 3 Equipment Request types the weapons and shields fall under:
This is very straight forward as very few equips created at a workshop require a specific focus. Equips with the Shielding/Parrying/Might/Power/Serpentis prefixes/suffix can only be used by units that are Combat focused. Equips with the Enchanted/Aura/Runic/Orphic/Crystallus prefixes/suffix can only be used by units that are Mystic focused. Almost everything not containing those prefixes/suffix can be requested with any focus. Accessory Requests follow similar request mechanisms.
It should be noted that the target unit you want to request a piece of equipment must have been in the active party and participated in 2+ battles before exiting to the World or City Map. They will have a 90% chance to request the target equip.
Remnants can only be used by specific units. They also have a high enough Equipment Rank that they will take precedence over everything else a particular unit can request as units will always attempt to request the highest ranked item.
It should be noted that unique leaders will not request items from outside their current upgrade path. Another quirk with requests concerns units that start in the Power Grip stance. If the unit can use the One-Handed stance with their currently equipped weapon (e.g.: Glenys and her M-sized Katana), they will default to the One-Handed stance after receiving their requested weapon.
Despite not being able to request equipment after exiting to a map, Soldiers can request equipment dropped after battle. It should also be noted that this can leave them with subpar equipment if they request an Auld/Antiquitis/Ancient variation as it will mark the end of an upgrade path.
If using the equipment tweak, there's a chance that it may affect some of their requests, such as locking them out of Remnant requests.
All the equips are sorted based on size and their internal game order.