You’re correct. They’re not closely related other than by name.
Wikipedia says:
The American robin (Turdus migratorius) is a migratory bird of the true thrush genus and Turdidae, the wider thrush family. It is named after the European robin[3] because of its reddish-orange breast, though the two species are not closely related, with the European robin belonging to the Old World flycatcher family.
Really? European robins look very different, the only similarity is the red breast (and even then, it’s only a superficial similarity)
Edit: I’m not trying to contradict you, just slightly surprised
You’re correct. They’re not closely related other than by name. Wikipedia says: