A URL would be more helpful than a var_dump.

Are you using a SEF/SEO extension?

First off, the links to tags should not generate a 404. If you have an items with a url /path/to/item, then your tags will inherit the item's path and become /path/to/tag/name_of_tag.

Now, I see you have a plugin to do multiple categories. Which one is it as it could be messing with the output. I do see duplicate entries in categories in "Published in...".

Additionally, try switching the URL suffix (.html) off. This could be borking the URL router.

Finally, create a menu item in a menu that you don't use in the site (a hidden menu, sort of) and point that menu item to all K2 Categories and use "content" as the menu alias. This menu link will make sure that your tags that have NO direct menu item always inherit the path /content/tag/tag_name.

I haven't tested K2 with Falang latest (didn't know it's even maintained to be honest). And frankly, I'm about to remove support for it as it's much simpler to do multi-lingual sites with core Joomla nowadays.

The extension you mention probably causes the issues. If you want a real, working solution for multiple categories, Yiannis from Web357.com launched this just yesterday: www.web357.com/joomla-extensions/multiple-categories-for-k2

It's really a solid solution for multiple categories in K2 and I've personally tested it. Make sure you completely uninstall "Additional Categories for K2" as it hasn't been updated for many years unfortunately.