Does Apple Watch Cellular clone the SIM card

No - the sim card only provides an ICCID number to the phone. That number is read once per boot as the phone just uses the sim to identify what account to bill for service when registering with the nearby cell towers (IMEI and ICCID together let the system know what number to assign your phone and who to bill).

Something else must be causing your observed behavior.

Keep in mind, there is no guarantee two phones in the same location are not on distinct cell towers. Especially if one tower is overloaded or doesn't have high speed data so the phone /carrier might pick a faster connection that happens to have less bars of signal.

What does your carrier say if you call them for a site survey / diagnostic?

This is the same set of steps you would use if you had a sim card, it's more about clearing the account that device has registered with the iMessage service.

Try to turn off iMessage completely, totally log out of the system in Settings > Messages. When you re-enable it, ensure you log in with your own Apple ID. At this point, you should be able to message to your registered email address.

If that doesn't work, you may need to back everything up and restore the device. This is somewhat problematic without a SIM card, since you may get stuck at activation if you don't at least have an inactive SIM from the proper carrier.