The customization options will be slowly unlocked as the story goes on.
For example, at one point Asterion could bring up that a previous master would prefer his horns in a different shape. The player is prompted to choose a different shape (or stick with them the way they are). Asterion will respond to your choice by getting happy, proud, sad, etc.
Some can be unlocked automatically, for example Asterion would be working out wearing a loincloth and the player will suggest some modern gym clothing.
Other accessories can be found by sending parties to explore the labyrinth or by interacting with different guests.
At the start of every day you'll have the option to open the wardrobe menu and choose Asterion's outfit/fur color/etc