Is there any plugin or extension that able to allow adding Joomla article under many categories?
By default, this is not available in Joomla, however you have 2 options:
Rather than assigning multiple categories to your article, you could use tags instead. Cause to be honest, this is kinda of what they are used for.
There is an extensions called CW Multicategories which does allow you to assign multiple categories to an article. Even though I have not tested this, the ratings and reviews are extremely good so I can only assume it will do exactly what you need it to do.
Hope this helps
I know the question is more than 1year old. But for those who would get here, I would like to provide some informations. CW Multicategories extension has been unpublished on JED at the moment. Reason is recreating to new generation of extension which offers more possibilities and better experience with the extension. Right now is being reviewed by JED admins, however it is taking them really long time (reaction time for replies is in weeks). You can get the extension still on developer's website - http://extensions.cesky-web.eu/extensions/cw-multicategories
Have a nice day