Is this one of the sleep alarms, or just a regular alarm? I just tested a regular alarm on my iPhone 14 Pro. I'm also wearing my Apple Watch Series 8. I set the alarm on the iPhone, and it went off on both devices, the Watch and the iPhone. The next test I did was to set the alarm on my iPhone and put the Watch on silence. That time, the sound went off on the iPhone, and the Watch vibrated.
If you go to the Watch app on the iPhone, you can change the setting for Push Alerts from iPhone. According to the explanation under the setting, it will push timers and alarms you have set on the iPhone to the Watch. So I took the time to set another alarm on the iPhone, took the Watch off silence, and turned off this switch. That time, the alarm only rang on the iPhone. I hope that is what you were looking for.