There are many reasons why the screw might not unscrew. None are really important since your only objective is to remove it.
The usual hex in the screw head is very similar to a Nobel, but there is a minor difference which (in this case) will probably not allow you to unscrew the screw (I suspect it will not fit closely enough to remove a tight screw).
I presume you have good vision of the head. Sometimes practitioners use different head screws (eg universal head) made by non genuine suppliers. If it was a universal head, it can be very hard to differentiate if you cannot easily see it. So my first comment is to get good vision of the actual screw head.
In my case (I inherit lots of patients with unknown implants and screws) I have about 30-40 different drivers which have been collected over 35 years. I simply try them individually until I find one that fits tightly enough to unscrew the screw. You probably do not have access to such a range of drivers but you might know a colleague who does. Perhaps borrow them.
Alternatively, I mentioned a few weeks ago about an Astra or Nobel (both are similar) screw removal “bur” which is latch grip and fits in a slow hand piece and can be “pushed” onto the screw head and then run the hand piece in reverse. A reduction hand piece works best due to slower revs and more torque. However this method might not work since it is designed for broken screws and therefore there would usually be more surface area for the bur to “grab” onto. However it is cheap and easy to get one of these burs so it would be high on my agenda to try before I went too much further.
The next step would be possibly a screw removal kit you mentioned. However these are expensive and not as easy to use as you might think.
Therefore, if you have time and a lot of old high speed burs, you might like to try to cut off the top of the existing implant crown to expose the offending screw head. Once exposed, there are a variety of possibilities. You might like to “slot” the screw head (with a high speed bur - careful it is not as easy as it sounds) and use a slot driver (rather than a hex driver). Or if you create a “groove” around the head, and you can grasp the head, you might be able to grasp the head with any type of forceps you have available (eg root or artery forceps) and twist it loose.
So you can see that there are many ways you could attempt to remove the screw. To some degree it is a little like extracting a root tip. You just have to think about different ways to get adequate access to the screw and try various instruments until you can get the right amount of purchase on it to remove it. I am sure you will find a way. We dentists are very inventive!