Child's Apple watch showing parent's name/contact info

Do your children each have their own Apple IDs?

Did you set up the watches as described in this article?

Set up Apple Watch for a family member - Apple Support

Sorry for being unclear.

If you change the "Me" information on their watches and add your phone numbers instead, your text will always show up as sent by you.

In the example mentioned above, the "Me" information on your daughters watch should contain your daughters phone number, and not yours.

"Me" always relates to the owner of the device/phone number.

Did you make sure, that in your contact list, your contact details are not saved under the name "Me"?

You could also try to set up another contact card for your name and share this contact information with those two Apple Watches, as explained in section #9 in the related article from above.

If you open the Contact App on their Apple Watch and tap their profile picture at the top right, does it show their phone number?

Contacts on Apple Watch - Apple Support

Maybe you should start the Family setup again, now that you made sure that your contact info is correct on your iPhone and in iCloud contacts.

You're welcome, helping other fellow users, that's what the Apple Community is here for. 🙂

Good Luck!