Magical Effects or Special Properties
One of the major things to consider are any magical effects or special properties the item possesses. Chapter 7 of the Dungeon Master’s Guide (DMG) focuses on treasure and rewards, including magical items.
Pages 141-143 detail special properties an item may possess. Whilst this section of the book is focusing on magical items, you as the DM could rule an item has one of these special properties but is not magical. For example, a helmet may have been designed for a cloud giant, making it too large for a medium sized creature to use. Plate armour that's not properly fitted may have the painful property. A shield may have an in-built compass or sundial.
Size and Shape
Following up from that, the size and shape of a weapon or piece of armour may be something to consider. Pages 140 and 141 of the DMG state, under "Wearing or Wielding Items", that:
In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build. Many magic garments are made to be easily adjustable, or the: magically adjust themselves to the wearer.
Rare exceptions exist. If the story suggests a good reason for an item to fit only creatures of a certain size or shape, you can rule that it doesn't adjust. For example, armor made by the drow might fit elves only. Dwarves might make item only useable by Dwarf-sized and dwarf-shaped characters.
When a nonhumanoid tries to wear an item, use your discretion as to whether the item functions as intended. A ring placed on a tentacle might work, but a yuan-ti with a snakelike tail instead of legs can't wear boots
Again, this is only considering magical items. A non-magical item will not will not magically adjust its size or shape (though an armour smith may be able to adjust the piece for you, for a fee). Below is a variant rule from page 144 of the Player’s Handbook which details how size may be important for armour:
In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won't fit in a halfling's leather armour, and a gnome would be swallowed up in a cloud giant's elegant robe.
The DM can impose more realism. For example, a suit of plate armour made for one human might not fit another one without significant alterations, and a guard's uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.
Using this variant, when adventurers find armour, clothing. and similar items that are made to be worn, they might need to visit an armour smith, tailor, leatherworker, or similar expert to make the item wearable. The cost for such work varies from 10 to 40 percent of the market price of the item. The DM can either roll 1d4 x 10 or determine the increase in cost based on the extent of the alterations required.
How It's Used
For armour, this seems fairly obvious, you wear it and gain the AC bonus. However, certain design features, told to the players when describing the armour, may warrant a special feature, as described above. For example, gauntlets with spikes on the knuckles may add +1 damage (piercing) to an unarmed strike.
A parrying dagger or sword breaker (it doesn’t actually break swords) may grant increased AC as they are used, mechanically speaking, like a shield. However, they would still have the 1d4 piercing damage a dagger does.
Finally, whilst not a hard-and-fast rule, you could make your items more interesting by attaching roleplay elements to them.
For example, a fun idea could be to offer the player’s a normal shield with a yellow square painted in the centre, taking inspiration from Heraldry and Abatements. Whilst the yellow square does not offer any mechanical advantages or disadvantages, in history, a person with a big yellow blotch on their shield was easily identified as a coward. This could lead to some unique roleplaying opportunities, such as people mocking the owner of the shield. Or, wearing armour bearing alliance to an enemy army, a cult of necromancers or the Thieves Guild may cause guards of a city to attack or arrest you. It could also be done in reverse, giving the player’s the weapon or armour that an ancient hero once owned, again, allowing for unique roleplaying opportunities.