What exactly is the problem?

This means that Firefox isn't blocking the Flash plugin on that web page, so Flash should be working without problems, probably because you previously allowed the Flash plugin.

If Firefox is blocking the Flash plugin then you would get a blue highlighted plugin icon on the location bar with the allow choice and to remember this allow choice.

You can inspect and manage the permissions for all domains on the about:permissions page or for the domain in the currently selected tab by clicking the "Site Identity Button" (globe/padlock) on the location bar and "More Information" to open "Page Info" and go to the Permissions tab.

• https://support.mozilla.org/kb/how-do-i-manage-website-permissions
• Tools > Page Info > Permissions

how can I remove the icon? Wasn't shown before and I want to hide or remove it, please.

You can't.
You should see an icon on the Navigation Toolbar and not the notification that the plugin is allowed if that is what you get.
Current Firefox versions are more strict and block all plugins by default and show an icon on the location bar.
Flash is an exception because this plugin is commonly used on a lot of web pages and the Flash plugin is enabled by default, but you still see a gray plugin icon on the location bar.
You can set the default on the Firefox/Tools > Add-ons > Plugins page.

I don't see the icon in Chrome, Explorer or the older Firefox versions.

I'd say: remove the thing devolopers! Next thing I'll do is search for an older Firefox version.