Currently, you can display files from different templates in the same section but not upload them. If you have a lot of file types in a page, you have to add a new section for each so the user can upload any of them.
The files section should support uploading with multiple file templates. The new uploader dialog is perfect for this. If only one file template per file type (e.g. image for jpeg, video for mp4) is available, templates should be assigned automatically. If multiple templates are available for one file type, a select similar to the Page blueprint/template selector should be display in the uploader dialog.