Is it asking to install apps or asking you to grant permissions to apps you have already installed?
In recent android versions the user can control the granting of many (but not all) permissions. You are asked to do this when an app first tries to use that permission. Some apps ask for all permissions they need when you first run them (and may shut down if any are refused, thereby putting you in the old position of having no control other than to accept all or not use the app), but others really do just ask when they try to use them. And some of those, when refused, will just make another request later. I've one app that's like that, and handled it by disabling notifications from it as described above, but I can imagine some cases where an app might stop at some point (e.g. if you deny it something it uses for authentication it will eventually refuse to run).
So I can't see FB installing apps (putting nags in its own app to install Messenger is something different), but I can imagine it asking repeatedly for a permission you may have denied it.
If it's actually the permission to install apps then it's obvious that certain Google apps will need that, but FB shouldn't and shouldn't ask.