Hello,
Thank you for the report, and for the analysis in your two messages of this afternoon: it is exactly right, and it is the one cause behind both points of your first message.
In 6.6.0 the "Main category" setting of the category and product menu items became a multi-select, so that a listing can be built on several categories. A menu item saved since that version stores its category as a list, even when you only picked one, and ten places in the code still read that setting as one value. A list read as one value gives the number 1, so a menu item pointing at exactly one category was no longer recognised as the page of that category.
That is what you are seeing. When HikaShop no longer recognises the menu item of a category, the category tiles of the parent listing stop pointing at the address of that menu item and fall back to the address HikaShop builds itself, /owen/izm instead of /owen/izmeriteli-regulyatory, or the category segments get added on top of the menu path. Both of those addresses carry the category in the URL, which is the case where HikaShop checks that the category is published, so an unpublished category answers "Category not found" through them. Your own menu items created before 6.6.0 still hold the old form of the setting, which is why the older categories kept working while the new ones did not.
No, we did not add a published check: the check on the category of the URL has been there since 2019, and it is deliberate, a category you unpublished should not be reachable by typing its address. Your workflow is not affected by it: the menu item you publish for that category remains a valid landing page and does not go through that check. The addresses which stopped honouring the menu item are what pushed you into it.
As for the reason for the multi-select: a listing menu item can now show the products of several branches of your catalog on one page, which was asked for regularly. The picker posts a list as a result, even with a single choice, and the places reading it had to follow, which we missed. A menu item naming one category is the page of that category, exactly as before; a menu item naming several is the page of none of them, so the router and the listings keep their previous behaviour there.
The fix reads the setting as the list it now is, everywhere. A menu item naming one category is the page of that category again, so the listings link to it, the router keeps the menu path alone rather than adding the category segments on top of it, and product links pick that menu item too.
Two more problems came out of the same reading and are fixed as well: a category listing menu item whose main category is emptied answered "Category not found" instead of showing the categories of the shop, and a product listing built on several categories printed an "Array to string conversion" notice.
Please download the install package of HikaShop again and install it on your site to get the patch. Your menu items need no change, they are read correctly as they are, and the ones you edited by hand in the database can simply be saved again from Joomla.