How can I prevent Siri from announcing who it's calling when I ask it to make a call?
You can't disable all Siri feedback but you should be able to cut-off most of the chatting. 

Source:


Go to ‘Settings -> Siri &amp; Search’.
Tap on ‘Voice Feedback’


You'll see 3 options to choose from:


  
  Always On: This is the default option. Siri will vocally respond to
  you after each query, no matter if you use the home button shortcut,
  "Hey Siri," or have your ring switch set to silent. If you're here to
  silent Siri, do not choose this option.
  Control with Ring Switch: When you set your ring switch to silent,
  Siri will not vocally respond to you, but there are caveats. Siri will
  still beep and respond vocally if you initiate a discussion using "Hey
  Siri" or if you're connected to a Bluetooth device, headphones, or
  CarPlay. So, the only time Siri will remain quiet is if you use the
  home button shortcut to activate her/him. If you unmute your device,
  Siri will again vocally respond in every situation.
  Hands-Free Only: Just like the previous option, Siri will only beep
  and respond vocally if you initiate a discussion using "Hey Siri" or
  you're connected to a Bluetooth device, headphones, or CarPlay. The
  only difference here is that you don't have to remember to set your
  ring switch to silent.
  


The author of that piece sums it up nicely:


  As you've probably noticed, there aren't really any great options for
  keeping Siri silent on your iPhone. The two not-always-on options
  listed above are virtually identical, only one requires you to set it
  manually each time using the ring switch while the other is automatic.
  
  So what's the answer to total silence, like when you're using "Hey
  Siri" but don't want voice feedback? Turn your volume down.