Same boat as me.
I have a Wireless bluetooth Xbox ONE S controller and it works fine with a £5 bluetooth dongle.
You must establish first whether your controller is actually bluetooth..
However. If you want to have more than one connected at the same time, you must buy an official xbox adapter.
Just be careful though as its really hard to tell even your pictures are not a 100% guide
My son has the same looking controller and its 2.4ghz wireless.
The 2.4ghz wireless one with coloured buttons is model 1697.
The bluetooth wireless one with coloured buttons is model 1708.
My red Xbox controller is 2.4ghz wireless no coloured buttons model 1708.
Says this in battery compartment so its not quite straighforward.
Also really annoyed as the new style XBox 2.4ghz wireless adapter (£19.99) is out of stock everywhere including MS (you can order but its put on pending). ETA I was told was end of Feb.
I didnt fancy a 3rd party one so I went on a very well known brown box site and bought one off their sold by company B fullfiled by company A and its a bloody fake.
So I reckon the ones you can find on another well known auction site might be the same. Looks like there are a lot of fake ones doing the rounds as the reviews say the same thing!
Below is fake one stuck in the USB port and below that is what a fake box looks like.
So I've ordered one of the 3rd party 2.4ghz Wireless USB adapters gets good reviews so will wait and see if that works.