No, many words can introduce nonrestrictive clauses. However, only "which", "who(m)", and "whose" typically introduce nonrestrictive relative clauses.
The relative pronoun "that" is normally used only for restrictive clauses, so the standard guidance would be to omit the comma in your example sentence. Most people would follow that guidance, and including a comma there would often be considered an error.
However, people have greatly varying opinions about commas, and they tend to be put to a much wider variety of uses than other punctuation marks. Therefore, many people would consider that comma to be perfectly acceptable.
My recommendation is to stick to the standard guidance in your own writing but to accept the fact that many heretics refuse to bow to what they consider the tyranny of the English Academy and that we all just have to learn to live peacefully side by side with one another.