That argument in and of itself doesn't cause any harm to anyone.
The harm comes when government institutionalizes marriage in a way that gives it greater rights or scope than someone in a civil union would have. In many states, this isn't the case.
Personally, I don't particularly care what the word marriage means, I'm not religious in any meaningful way. But arguing that the simple inability to use the word marriage does harm to a person is a pretty weak argument in my mind.
Further, your analogy is wrong. From his perspective, imagine you always played soccer with a round ball, and couldn't use your hands. Suddenly a group of people show up with oval spheroid and claim you should be able to use your hands. They then say that not letting them call their sport soccer is unfair and harmful. But ya know, we don't call that sport soccer, because it's different, we call it football.
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As an aside, I think the proper way to handle the marriage debate would have been to remove any and all rights associated with the word, and force all couples to get a civil union. Tax that instead. Churches are welcome to marry people all they desire (on both sides, gay or straight as they please according to their beliefs)