I’m not sure if I understand your request.

What you can do is just the totally basic style element or attribute:

  <svg …>
    <style>path { fill: red }</style>
    <path d="…" style="stroke: green" />
  </svg>

But I’m not saying you can apply styles to the inside of an image from outside; that, as I have remarked, is a different matter, about heritability—you can’t do that with any technique but inline SVG. Not <img>, not <iframe>, because it’s not cross-document.