Sometimes when you're switching between SIMs on an iPhone or iPad, your iMessage settings need to be reset in order to work properly.

• Tap "Settings"
• Tap "Messages"
• Toggle "iMessage" to OFF and wait a second
• Toggle "iMessage" back to ON
• You'll see it says "Waiting for activation" 
• Tap "Send & Receive"
• Make sure your Apple ID email address is checked off

Selecting your email address will allow you to continue to send and receive iMessages using your brand-new eSIM data connection, without relying only on your phone number.