You are right. The phrasing "calling him names" and others that are similar means being derogatory or saying negative things about others. Since that's a colloquial term, I don't know how it got started.
Your examples of good expressions are correct, too, except I would not use capital letters for "strong" or "prince."
1. Don't call me names.
2. He is calling me a name.
3. She is calling him names.
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Are they all grammatical?
Which ones are commonly used?
Does 'call somebody names' mean 'speak ill of others'? Why is that?
We can see good expressions as follows?
He calls me Bob.
He calls me Strong.
He calls me Prince.
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