Can I make Chrome or Mozilla have a favicon for local files like PDF or similar?

The thing is that there is just the generic favicon of a faviconless page. Any solution is valid, like adding html to the loaded file or configs of the browser.

Solution

You could create a HTML page which has a favicon and just add an iframe to it. Although this may not be what you wanted but in this example it would show a .png file with the stackoverflow icon.

body {
  margin: 0px;
  padding: 0px;
}

* > iframe.maxsize {
  position: fixed;
  width: 100%;
  height: 100%;
}

.maxsize {
  display: block;
  width: 100%;
  height: 100%;
  border: none;
}

<!DOCTYPE html>
<html lang="en">

<head>
  <meta charset="UTF-8">

  <title>My File</title>

  <link href="https://www.google.com/s2/favicons?domain=stackoverflow.com" rel="icon">
</head>

<body>
  <iframe src="https://www.google.com/images/branding/googlelogo/2x/googlelogo_color_272x92dp.png" class="maxsize">
        Your browser does not support iframes!
    </iframe>
</body>

</html>