As far as I have seen with my limited exposure to only fairly recent wireless mice, working or malfunctioning, you're right - if the wireless receiver / transmitter in the USB port is being detected properly, a wireless mouse model shows up in Device Manager even if the mouse isn't actually working.
Is the subject mouse wireless ?
If yes, try plugging in the wireless receiver / transmitter for it into a different USB port.
Troubleshooting USB device problems including for flash drives, external drives, memory cards.
See Response 1: