Most home users will be assigned dynamic IP addresses by their ISPs. The ISP is allocated a pool of addresses by the Internet authorities, and assigns these to their customers when their routers, or any device connecting in place of the router, goes online.
There is a protocol called DHCP that allows the ISP to configure the addresses that a customer needs to access the Internet through the ISP. The same protocol can configure addresses for devices connecting to a home router. Using DHCP makes the set-up process plug-and-play and eliminates many of the problems that can occur if people have to configure addresses manually (as they are easy to get wrong).
With DHCP, each connection has a lease time. Once half the lease time has expired, then the connected device can start requesting a renewal of the lease. This can mean that you may stay with the same public IP address for many weeks if you keep your router permanently switched on and connected to the ISP. Theoretically, if the device stops using the connection, but resumes within the remaining lease time, then they should always be assigned the same address again. However, many ISPs detect the loss of the low level connection and terminate the current lease at that time. At home, you might use a PC in the evening with a specific address assigned by the router, and get the same address when it is booted up the following morning if the lease issued by the router is still valid.
Websites do not store IP addresses beyond the current working session. They rely on several parameters to identify the connection. One is the IP address (obviously), one is the port number, and one is the session ID. If two people on two computers in your home access the same website, the website will see only one public IP address, however, the two sessions will be using different port numbers and different session IDs.
If you open two tabs in the same Internet browser, and log on to a website from one tab, then you will normally have an open login on the second tab. This is because the browser will have the same session ID for both tabs, and the website will 'know' there is a valid login open for the IP and session ID although the port numbers are different. The port numbers have to be different to allow information to be directed back to the correct tab.
If you open two connections from the same computer but using two separate browsers they will have separate session IDs and the website will require a login for both sessions.
Changing your router may or may not mean you end up with a different IP address. If you ISP assigns a static (fixed) IP address, then it should not change. If it could change, then the ISP could randomly assign the same IP to your connection. However, if the lease time of the previous connection has not expired, when the new router requests an address (using DHCP), then the ISP will normally assign a different address. This is because the ISP recognises the router by its hardware MAC address, and will see it as a different device.
Finally, if your ISP assigns a dynamic (changeable) address and you need to accept incoming connections because you are hosting a website or game server, then look for a DDNS service (Google it). These allow you to have a domain name that tracks the changes in your public IP address. Internal IP addresses between your router and your server machine are handled by fixing the server on an internal static address and port forwarding in the router.
I hope this helps.