Commas don't always have anything to do with grammar.
These two commas are filling different roles. They both indicate pauses in speech, of course, but the first one (arguably) has a grammatical role. It is differentiating between coordinate and cumulative adjectives. Coordinate adjectives use the comma, and they each separately modify the noun - the story would be unique, and it would be their own. Cumulative adjectives, which don't have commas and usually obey the so-called "royal order of adjectives" (I have no idea when or why the royal was introduced to that term), well, accumulate rather than applying separately. One interpretation is that the one nearest the noun applies to the noun, forming a noun phrase, and then the next one out applies to that noun phrase, and so on.
The second one is stylistic. It is not required, but it helps readability - just as the equivalent pause would help people hearing it read aloud to understand - by indicating that you are back in the normal flow of the sentence. This isn't like a parenthetical clause, because there's no separate clause at all, and so there's no requirement for the comma. However, like many usual stylistic choices it's strongly recommended because it actually makes the writing more understandable, reducing the cognitive burden on the reader.