Unable to activate eSIM on my new Apple Watch Series 10

In addition to what Ingo2711 posted, there is also the Apple Watch Family Plan

Set up Apple Watch for a family member - Apple Support

Set up Apple Watch For Your Kids - Apple Support

Some limitations: Cannot receive SMS/MMS messages, as that depends on a fully paired iPhone. Messaging is pretty much limited to Apple's Messages, unless the messaging service provides an Apple Watch app. Alarms, alerts, notifications on the iPhone are not sent to the Apple Watch. Apple Pay via a credit card is not available. Also not available: Medications, respiratory rate, irregular heart rhythm notifications, ECG, AFib History, Cycle Tracking, Sleep, Wrist Temperature, Blood Oxygen, Walking Steadiness, Audiobooks, Camera Remote, News, Shortcuts, and the double tap gesture.

Did you follow the steps in this related article to set up your cellular plan for your watch?

Set up cellular on Apple Watch - Apple Support