It is possible to add custom media size in admin media edit?

After looking into this I think the answer is no.

It looks like the area was built as a manner to preserve the thumbnail, or alter the thumbnail (which has a specific use case in WordPress), when making edits to an image. That's why medium, full, and large don't appear here as options. The point is "if you're messing with your image, how do you want to handle the thumbnail?"

If you need a custom thumbnail image size, you're better off editing the thumbnail and then you can use this area to control the impact on that when editing images. I don't think extending this to repurpose it for "choose your image size to edit" is something you can do without a more complex plugin.

Steps to change default thumbnail size:

1. Go to your WordPress dashboard.
2. Navigate to the Settings > Media tab.
3. Look for the Image sizes > Thumbnail size section.
4. Change the default width and height of your thumbnails in pixels.

add_image_size( 'medium-something', 480, 480 );

// Register the useful image size for use in Add Media modal
add_filter( 'image_size_names_choose', 'your_custom_sizes' );
function your_custom_sizes( $sizes ) {
    return array_merge( $sizes, array(
        'medium-something' => __( 'Medium Something' ),
    ) );
}

Be aware that if the size does not already exist for an image, it will not be an available option in the dropdown. New image sizes are applied only to new uploads without regenerating thumbnails for all existing images.