Matthew Arnold was (and is) well known not just as a poet, but as a literary critic. So it stands to reason that he takes poems, and their power, Very Seriously. The reference to Sophocles in "Dover Beach" also shows the fingerprints of a guy who cared a lot about education and classical learning.
Still, we think what really marks this as a Matthew Arnold poem though is that dark undertone. We see that "eternal note of sadness" (14) in a lot of his poems. Not that he wasn't capable of appreciating beauty—we see that all over the place in this poem, too. But the general turn toward sadness and loss is a major feature of a lot of Arnold's poetry and of his outlook on the modern world.