No, it shouldn't. At least: It doesn't. The computer icon resolves to a virtual folder and XY doesn't support a registration for it atm.
But there are so many ways to invoke XY and to get to this place.
One way:
Create a shortcut (.lnk) file for XYplorer.exe
Right click that, use the "Properties" context menu entry.
Let's say the target line shows this:
D:\Users\Highend\Tools\XYplorer\XYplorer.exe
Change it to:
D:\Users\Highend\Tools\XYplorer\XYplorer.exe "Computer"
Ofc you have to use the path that's valid on your system, not on mine
Pin that shortcut to the taskbar. Depending on it's position you can invoke it via a single left click or the Win key + <position on the task bar>
E.g. if it's the second item, it would be: Win + 2.