Hello,
Edition of characteristics is different than edition the "product / characteristic".
We can see that under "product / edit" and "product / new", you don't have all fields, which means that we can't say if your vendor can or cannot edit characteristics/variants on products.
But since you're saying that you don't have the same interface than our demo website, we can conclude that the ACL are not configured accordingly.
Regards,
Jerome - Obsidev.com
HikaMarket & HikaSerial developer / HikaShop core dev team.
Also helping the HikaShop support team when having some time or couldn't sleep.
By the way, do not send me private message, use the "contact us" form instead.