The problem here is that you did not import IonTextarea from @ionic/vue and provide it to your Vue component, so you are not able to use the v-model binding.
If you notice in your browser’s dev tools console, the following warning is there:
[Vue warn]: Failed to resolve component: ion-textarea
It is a bit confusing since the ion-textarea component does render, but what you are seeing is the regular Web Component, which does not have any Vue support. By importing IonTextarea you are importing the Vue wrapper which brings v-model support.
I recommend checking out our Ionic Vue Troubleshooting Guide as it covers other common issues developers may run into.