There is this persistent myth that it is, under the GDPR, not allowed to use a noreply email address when you mail someone. But this isn’t so. The GDPR doesn’t mention email a single time. It is mentioned one time in consideration 23 but that paragraph is about preventing that a natural citizen gets deprived of his rights under the GDPR in those cases that a controller or processor resides/works outside of the Union.
Apart from the GDPR it still can be possible that national legislators have forbidden it.