inventory is carried in the player's backpack, which is carried around by Leo when he is in your party. It has an unlimited capacity for items and beards.
At the start of the game, the player has no backpack and can therefore not permanently pick up any items. After the tutorial Leo joins the party and the bag becomes available, allowing the player to pick up and carry eligible items.
Beards are a special type of item. They can can be seen in the BeardPad device under the Beards tab. They can also be seen when changing your beard which is done via the main HUD display. There are 23 beards to collect.