Connecting router to modem that has wifi already built in

I am planning to use a router which is connected to the modem that already has an router built in.

The reason for that is I am living with my friends. They all have their own pcs laptop and everything connected to the modem with built in wifi. We have 2.5 and 5 ghz as well as guest wifis from the same modem-router thing. Only one box.

I am into online business and I do not want my accounts and websites to have problems(blocking, suspension, fines)just because they have their own amazon seller account or they are doing some bad shit.

Will connecting a router to a modem with built in wifi work or I need something else? Do they share the same IP? Can websites like amazon understand we are using the same ISP and have multiple accounts at home?