Does an Apple Watch use the same phone number as your Apple IPhone?

Yes.

Assuming you buy a cellular model, AND your iPhone carrier supports the iPhone, AND you decide to activate the cellular radio, then your carrier will add about $10 per month to you iPhone cellular bill and you iPhone phone number will be shared with your Apple Watch.

To clarify on the previous answer I'll add that your  Watch will be assigned a different number by your carrier. But this is only for use by the carrier to keep it separate from your iPhone when it comes to billing. The  Watch will answer the same number used by your iPhone..