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Yes, it does appear as an extension.

Even before I looked at your screenshot I can tell you that the assertion that it does not appear as an extension is completely false. If that was the case, Joomla could not find updates for it (it only finds updates for installed extensions). It found an update and you tried to install it, therefore it is an extension.

Your screenshot of the Extensions Manage page excerpt listing FEF as an extension of the type File is the final and most conclusive proof that Joomla does indeed show it as an extension! You just click that checkbox at the left side of your screenshot and then look up at the toolbar, click on the Uninstall button.

Remember that the seven Joomla extension types are: Component, Module, Plugin, Template, Package, Library and File. The first four are easy to understand. A Package extension is a collection of related extensions, any number of any combination of the other six extension types which are installed, upgraded and uninstalled together. A Library extension is a special extension which can only put files in the /libraries folder of Joomla but also does not support the full set of pre-update and post-uninstall actions allowed by other extension types, making it useless for anything except third party libraries shipped with Joomla. The File extension allows the placement of specific files in any folder of the Joomla installation. It's typically used for Joomla and extension language files (which you probably don't know given the market you're in) but nothing stops us from using to to add files to the /media folder — what FEF does. All of these are extension files, all of these have XML manifests, all of these can be uninstalled from the extensions manager (except for the ones with a Protected status, used by Joomla core extensions to prevent accidentally bricking your site).