Your router is using an IP address assigned to you by your ISP. This IP address is public, it is the way web sites transmit data to your computer. If you had no IP address, you have no internet. The only way to hide your IP from other sites it to set your router up through a proxy server but it often makes your internet speed much slower as there would then be a middle-man between you and the internet.
*normal data transmission:
(you = your pc + ISP)
you > web page > you
*proxy transmission:
you > proxy > web page > proxy > you
*very much simplified
