Formatting the laptop would have no effect on the modem/router and should have no effect on other devices that are connecting to the modem/router.
Now it is possible that there is an issue with your modem/router. Have you tried rebooting it? (powering it off and back on). Also, have you checked your connection? Any issues with the connection (and it dropping) can cause the problem you are having.
Also, it could be the number of devices connecting to the modem/router. There are settings within most of them that will limit how many devices can be used (you can usually turn that off and on) or even add specific devices to block.
Lastly, depending on who your carrier is, it is possible that they have blocked that device on their end. If you have done anything that could be considered against their policies, then they could have blocked that device from using the WiFi. Or if the WiFi is owned by/controlled by someone else, the same thing is possible.