I installed the latest Arduino IDE on both, a Win XP computer and a Win 10 laptop, without connecting the board itself.
On the XP, under Tools, appear 2 COM ports, but on the laptop not.
You need to connect your Arduino board to the laptop using the USB cable.
If the Arduino board is not connected, any COM ports shown in the tools -> port menu are unrelated to Arduino (and selecting them when attempting to upload will fail). Likely the old computer has two built in serial ports; this was common 15+ years ago in the heyday of XP.
You must connect the Arduino board (and install drivers, if required) before any serial port will appear in the menu.