Will my family members get the same apps I download?

My mom and brother and I are getting a shared data plan for our new iPhones, and I've heard that there's something about everyone getting each app that anyone in the plan downloads. Is that true? I don't want to have to have all of the games my brother will download, and I don't want my mom having all the social media apps I download lol

No, the data plan has nothing to do with the apps. The apps you have access to depend on the Apple ID you use to set up the phone. If more than one person is using the same Apple ID on his or her iPhone, then they will all have access to the same apps, although they aren't obligated to download them all. Any app purchased with a given Apple ID can be downloaded by any device that is associated with that Apple ID, up to a limit of ten mobile devices.

So if you are not using the same Apple ID as your mom and brother, you won't be using the same apps, even if you share a data plan. If you are using the same Apple ID, apps purchased for any of the phones will be available to all of them, but you don't actually have to download every app you purchase on every phone, that's up to each of you to decide.