According to Apple's documentation (below) and my experience on the iPhone 14 Pro with silent mode enabled music should still play. Ensure your volume is still turned up when listening to music as if you have silent mode enabled and volume on zero your device will make no noises.
If you still encounter this issue I would highly suggest reporting a bug to Apple directly (see below) as this is not an intended feature change for the iPhone 14 Pro.
Feedback - iPhone - Apple
Turn Ring/Silent mode on or off on your iPhone - How to put your iPhone on vibrate, ring or silent mode – Apple Support (AU)
In Ring mode, you'll hear ringtones and alerts. In Silent mode, you won't — but your iPhone can still play sounds, such as when you play music or videos.
Use Ring mode
To put your iPhone in Ring mode, move the switch so that orange isn't being displayed.
Use Silent mode
To put your iPhone in Silent mode, move the switch so that orange is being displayed.