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Famous people always get some harassment. We used to call it "hate mail", now it's "hate replies". It comes along with being well known. Also, you can turn off notifications for @ replies from people you don't follow if it's an issue.

Facebook should add a privacy setting that allows you to disable @mentions completely, or limit them to people you have added as a friend. I would not be surprised if this is already a feature.

It does.

I was wondering about the @ thing. There is a strange account that keeps @ my account name and a few other people in my field whose names I recognize. I would like to be able to disable that.

You can block that on both platforms AFAIR.