Media and Site assets are separate libraries. The field you are editing determines which library and file formats you can select.
Media
Use Media for product galleries, brand and vendor images, category and collection images, blog featured images, author avatars, image attribute options and compatible custom data fields.
Site assets
Use Online Store > Site assets for favicons, theme section images, theme settings, SEO social images and checkout branding.
Reusing the same source
You cannot move a file from one library to the other. Upload a copy to the second library when you need it there. Each copy has its own uses; deleting one copy does not delete the other.
Access and storage
Your team needs view access to browse the relevant library and edit access to upload, rename or delete its files. Access to a product or page alone does not grant access to its file library. A missing action may reflect your permissions.
Library permissions control who can manage files in Store Admin and the app. They do not make a copied storefront file URL private.
Both libraries draw from the same storage allowance. Selecting a file again uses no additional storage; uploading a second copy does. See Storage and upload limits.