They are not leeching clout off anyone. And they are not using another entity's name.
They _are_ Amazon Employees. They are in no way factually incorrect by calling their group "Amazon Employees in support of <insert topic here>" and are entitled to call themselves that, because that is who they are.
Just because the words "Amazon Employees" contains the word "Amazon", that doesn't mean they are not allowed to use it. If Amazon would prefer them not to call themselves "Amazon Employees", Amazon is free to not employ them in the first place. Now that Amazon did employ them, it has to allow them to use the objectively correct way to describe their group. And apparently, if Amazon would like to stop employing them, it gets to pay a settlement to them or explain it in a court, because there are rules against that.