There are multiple pieces mixed up in your question:
1.) Every song has a filename, like My beautiful named song.mp3
2.) If you load this song in Mp3tag you can import a copy of this filename in the metadata tags, ie in the tag TITLE. You can use the menu Convert -> Filename - Tag (or press ALT + 2) and the preview of this converter let you see that NO extension like *.mp3 will be converted.
3.) If you already have the extension .mp3 in your TITLE tag, you can do step #2 again.
4.) If your content in TITLE is not exactly the same as the filename without extension, you can use a replace action to replace the unwanted part in TITLE with nothing = remove it.
5.) During a download of a song from an external source, you can not modify the metadata tags. You can only modify a completely downloaded local copy of such a song.
6.) Your player software can read the filename and/or the manually added metadata tags. This heavily depends on the various available player software.