My laptop apparently doesn't support USB 3.0 because the Elgato Game Capture HD60S shows me exactly this. Isn't there any adapter or an extension cable that you insert into the 2.0 slot and then the 3.0 signal of the Elgato Game Capture is recognized?
No, there's no adapter.
You also can't convert a slow USB 2.0 port into a fast USB 3.0 port.
Give Google a "USB to LAN adapter".
No.
It is impossible to achieve a higher interface speed using an adapter.
It's like trying to get a thousand out of a parking garage with a hundred cars by widening the exits.
What use is a USB-to-LAN adapter if the necessary USB port is not available?
If you have a USB-3 device and a computer that only provides USB-2 or no USB at all, but a LAN connection, then something like this can help.
This assumes that the device in question has an active USB 3.x hub and could supply the entire system with power, which is not the case with the Elgato Game Capture HD60S.
Sometimes that would be a different matter. The questioner stated, however, that his Elgato is USB-3 capable. Otherwise there are also active adapters that fix possible problems with the power supply.