Short answer
For a travel eSIM to work, it usually must be selected for Mobile Data and Data Roaming must be ON for that eSIM line.
On iPhone
Open Settings → Cellular / Mobile Data / Mobile Service.
Under Cellular Data, choose your travel eSIM.
Open the travel eSIM line.
Turn Data Roaming ON.
On Android
Open Settings → Connections / Network & Internet → SIM Manager / SIMs / Mobile Network.
Choose the travel eSIM as the line for Mobile Data.
Open the eSIM settings.
Turn Roaming ON.
Important
Roaming for the travel eSIM is expected and usually required. To avoid charges on your main SIM, keep roaming off on your main SIM.