Having the right element selected on the HTML DOM tree, right-click on it and choose “Copy” > “Copy styles”.

Source: https://getcssscan.com/blog/how-to-inspect-copy-element-css

You can use Clipboard.write() with both text and html parts:

TS Playground

// This will return the raw HTML, but perhaps you want to do something different,
// for example: recursively embed computed styles:
// https://developer.mozilla.org/en-US/docs/Web/API/Window/getComputedStyle
function serializeElement (element: Element): string {
  return element.outerHTML;
}

async function copyTextWithStyles (element: Element): Promise<void> {
  const html = serializeElement(element);
  const htmlBlob = new Blob([html], {type: 'text/html'});

  const text = element.textContent ?? '';
  const textBlob = new Blob([text], {type: 'text/plain'});

  const clipboardItem = new ClipboardItem({
      [htmlBlob.type]: htmlBlob,
      [textBlob.type]: textBlob,
  });

  return navigator.clipboard.write([clipboardItem]);
}