Hello,
What removes the videos is not Joomla but a HikaShop setting: when a product is saved, HikaShop filters the HTML of its description and removes the iframe tags, which is how YouTube videos are inserted. That's why it works in articles and not in products, whatever you change in the Joomla text filters or in TinyMCE.
Go to System>Configuration, in the Main tab, and in the Product section, set the "Filter HTML in description" setting to "No" and save. Then edit your product, insert the video again and save it: the video should stay this time.