As a non-native speaker “an” sounds better to me too. “Auf” seems more appropriate for an object fully resting on top of something, while “an” suggests something leaning aganst, mounted on, or hanging a vertical surface like a wall; or hanging from something like a clothesline.
But then, the way I hang my clothes is to wrap them over the clothesline, so they’re objectively hanging on top of the line, which suggests “auf der Leine”. But anyway, nobody here would bat an eye at either of those prepositions.
Even so, although you drape the clothes over the line, they’re not on it in the same way as a book on a table. They’re hanging there. An seems to me almost to imply some separation, or the concept of approaching but not quite being fully “on” the supporting object. (Or concept: this reminds me of all those verbs with the an prefix, which imply that the agent is approaching but not quite accomplishing the action described in the root.)
To me, non-native speaker, if someone used “Wäsche an der Wäscheleine” in part of a sentence, it would be as part of a temporal statement or an indirect command, or just indirect statement generally. “Auf der” would be a wholly nonjudgemental statement about the location of the Wäsche. I am not an educated person, but this is my initial reaction to the question. (I think of German an- like Latin ad-, and German auf like Latin in, except they’re not, the latter especially, cause translation is impossible.)
As a native speaker I agree that both prepositions are fine and there is no difference in meaning or tone. I would rather use “auf”. At the same time, I find that “an” is technically more correct for the reasons @Spectre has given.