The workaround is to upload the icons through the WP Media Library, but since the WP File picker usually has an upload tab integrated, it would be much better to have that integrated here, too.
My mistake. SVG Upload was not enabled in the Bricks settings on my test environment. It was somehow still possible to upload the first SVG (there is a big upload button in the middle of the WP File Picker when no SVG was uploaded yet). Therefore I thought SVG uploads are enabled for the user.