It truly is weird that some version of her iconic look has endured all this time. The kind of hilarious part about it is that I feel absolutely sure the outfit was meant to be a one-off joke.
You know, she was introduced during a storyline where the X-Men had been separated after their stint hiding in Australia, and after helping Wolverine escape being tortured by Reavers, she helps him heal up and becomes his "sidekick" as he finds and rescues newly-ninjafied Psylocke and I think they all start looking for the rest together. Anyway, I think it was just after they got Psylocke that she first appeared in the outfit -- just after or just before, anyway -- and the joke of it was kind of lost in time, because with her being the Robin to Wolverine's Batman, her yellow-trenchcoat and shorts look was a pretty obvious way of poking fun at the traditional yellow cape/green trunks Robin costume, which everyone had already come to agree looked pretty ridiculous, by that point.
The joke got lost somewhat though because over in DC, Tim Drake was introduced not long after Jubilee had popped up in Marvel, and of course for Tim, they'd designed and updated, more badass-looking version of the Robin costume, which has pretty much been the basis for Robin's look, ever since. So yah, her original look was one of those inter-company rivalry things that was kind of funny at the time, but made an obsolete joke almost immediately, by Robin's costume no longer looking like that.
As for why she still has remnants of that outfit, today -- I don't know, because people recognize it, I guess? Why does Luke Cage wear a yellow t-shirt?
Anyway, random comics trivia!
(Edit: Honestly, when I google "Robin original costume" and "Jubilee original costume", I still think it was pretty funny, for a passing visual joke. Just no way of knowing what's going to stick with any given character, though.)