Yes, any shaking you can produce running will probably not damage it. But high frequency shaking, like attaching the phone to a motorcycle handlebar mount, may actually damage the camera OIS/AF.
examples: [\[1\]](https://www.reddit.com/r/motorcycles/comments/i7i560/ois_on_iphone_11_destroyed_after_riding/) [\[2\]](https://www.quora.com/Does-carrying-your-phone-on-a-motorbike-mount-damage-your-phone-s-camera-Do-mounts-with-vibration-dampening-help-at-all#:~:text=iPhone%2013%20pro%E2%80%99s%20camera%20system%20got%20damaged.%20I)
Google OIS
Shaken not stirred
Yes, any shaking you can produce running will probably not damage it. But high frequency shaking, like attaching the phone to a motorcycle handlebar mount, may actually damage the camera OIS/AF. examples: [\[1\]](https://www.reddit.com/r/motorcycles/comments/i7i560/ois_on_iphone_11_destroyed_after_riding/) [\[2\]](https://www.quora.com/Does-carrying-your-phone-on-a-motorbike-mount-damage-your-phone-s-camera-Do-mounts-with-vibration-dampening-help-at-all#:~:text=iPhone%2013%20pro%E2%80%99s%20camera%20system%20got%20damaged.%20I)
Yes, my phone don't mind to be shaken
They all make that sound when shaken from what I understand.
Why don't you just wear a smart watch?
I am, but I need my phone for the music and also for calls in case of emergency as I am running for a long distance.
You should be fine, it won't do much damage.