A friend is a mutant fan so I need this answered (I would never use him)

"Playing to the Crowd: Modify William Stryker's attack value by the number of adjacent characters named "Purifier". If the target of his attack possess the Brotherhood of Mutants or X-Men keyword or team ability, modify his damage value by the same amount."

Legit triggers of damage bonus
Brotherhood of Mutants keyword
Brotherhood of Mutants team ability
X-Men keyword
X-Men team ability

Correct?

That is correct. (Keep in mind you still need adjacent Purifiers to make this work.)

Note the use of "possess" in there.

If my Spider-Man is copying Brotherhood, you would not get the damage bonus.

The trait, “Mutants are the Enemy”, ensures Spider-man will NOT copy the Brotherhood TA regardless