Translation:He only has one jacket and he wears it all the time.
Yes, Elfriede3001 is right. It must have been the any/and typo that (annoyingly) caused the error. "He has only one jacket" was accepted.
I agree that context should indicate, but a single sentence doesn't provide sufficient context. If the context is "I have a rain jacket and a windbreaker, ... " then "he only has one jacket" makes sense. But if the context is "I have a parka and a rain jacket and a sweater and a fleece" then "he only has a jacket" makes more sense.
It is confusing but I think Duo is making the point that the verb tener does not take the indefinite object so in this case it makes sense to translate it as one jacket instead of a jacket.
I don't think there's any reason why "tener" can't work with an indefinite object.
"I have a jacket" -> "Tengo una chaqueta/campera"
"interchangeable" is a pretty strong word
I think CheriStead has laid out pretty well the different contexts in which one or the other makes sense for the first half of this sentence, for example. Certainly either could be used in the sentence "He only has [...] jacket," but the foregoing context can determine which one fits.
Of course, there are also trivial examples: "He has two jackets. I only have one."
This woman is really hard to understand. I had to play the recording over and over. Could only understand her speech on slow play! Terrible!