There is no such thing as an extension cord/power strip that is approved for Royal Caribbean.
There are two reasons cruise lines don't allow extension cords/power strips. First, because if it has a surge protector, that is a fire hazard due to the way electricity works on a cruise ship. Second, because if it is too low of a gauge of wire and you try to pull too much power over it, that will cause heat and once again, that is a fire hazard.
Some cruise lines specify extension cords/power strips are not allowed as long as they don't have surge protection, as that is the primary concern. So when you see something advertised as cruise approved, what is really being said is that the product doesn't have a surge protector, thus meeting the requirements of lines than specify no surge protector. You are meant to think it is allowed on all cruise lines, but that is not what it is actually saying. It means it is approved for the cruise lines that allow those sorts of extension cords/power strips. Royal isn't one of them.
However, Royal's official rules are not strictly enforced. Security VERY seldomly will confiscate electrical stuff when the bags come on board. So ultimately, it's all based on your room steward. If they got and follow their exact training, any extension cord/power strip, even those advertised as "Cruise Approved," will be confiscated. Others may not know or care. Others may be swayed by something saying/showing as "Cruise Approved." It's impossible to know what yours will do until you are on board, though most seem to do nothing. This is why you get such a wide variety of feedback and responses. People buy stuff that says "Cruise approved" and have stewards not take it away, so they believe it's allowed, when in fact it isn't, the rules just aren't often enforced. While others will say theirs was confiscated, because they ran into someone who does enforce the official rules.