Profile banner customization is a Nitro-exclusive feature. If you cannot change your profile banner on Discord, then you are not subscribed to Discord Nitro. How do you get a banner on Discord?

From there, you can write a 190-character blurb about yourself (which can include markdown and links if you want), and Nitro subscribers can add a banner image in the form of a PNG, JPG ...

The top Nitro tier grants users access to animated avatars, custom profile banners, Discord Tags, emojis, sounds, stickers, two free Boosts, higher screen share quality, larger upload sizes ...