A customer has recently asked if they could use SVG images as the notification icon. Unfortunately the answer is no at the moment because of browser support.
I've just tested with an SVG image and Firefox (v54) works properly, while Chrome (v58, both on Desktop and Android) doesn't display it. So the answer is no, at the moment you can't use SVGs.
If you use an image format which is not supported (or an invalid image), the notification is still displayed on all browsers. The only problem is that the image will not be displayed on some browsers.