Nothing since you don't have Google Cloud Messaging on iOS, but it's a different situation. IOS is not open source software like Android (AOSP) is. If you want, you can have all the software on your phone be open source (save perhaps for drivers), even though most people will opt to install at least the Google Play Store, which requires you to install the whole google suite (or at least it used to when I flashed Cyanogenmod).

So anyone wanting to have a phone with open source products on it for security reasons, they totally can on Android, but it's impossible with iPhones. Signal probably relies on Apple's variant of Google Cloud Messaging, but since you'll always have that on your phone anyway, it makes no difference.