I just installed the latest packaged version (2.1.40) from the Anki website on my Fedora KDE desktop. I can start Anki from the terminal but not from the icon in the app launcher. If I click on the icon it appears and then disappears from the status bar without actually starting the app.
Maybe try the official version from https://apps.ankiweb.net/?
Please see:
https://faqs.ankiweb.net/third-party-linux-packages-and-source-builds-are-not-supported.html
I am using the official version. Sorry if I wasn’t clear enough in my post.
Oh, sorry for misreading your post. You said you installed it from the Anki website. I saw “packaged” and my mind filled in the rest 
Maybe the launcher has a problem? Try creating a new launcher or shortcut to the Anki executable. (I don’t know how to do that in KDE).
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So I can start anki from both:
~/.local/share/applications/anki.desktop
/usr/local/share/applications/anki.desktop
But I can’t create working links or copies from either of these.
Edit: Actually, it only works if I navigate to these directories in the terminal.
Not sure what the problem might be - maybe people on your distro’s forums will have an idea?