Lightning and Cloud have the same chin, meanwhile their personalities are far from similar. Cloud often held up a facade of bravery and strong-headedness while he was actually an insecure amnesia-filled guy. Not that I don't like Cloud, he's awesome.
Meanwhile, Ashe and Lightning have far more in common. They both don't take crap from anyone, and have extremely little patience with those that get in their way. For Ashe, it was Vaan and Basch, even, in the beginning that got in her way and annoyed her to no end, and for Lightning, it was... everybody. What I mean is, both Ashe and Lightning had their beginning period of cold-hearted focus and seriousness, and both eventually open up to all of the characters that bother them. Cloud was open to his female duo(Tifa and Aerith) from the beginning, lol.
Another reason, is that Ashe was relatively light-spirited and "Yuna-like" pre-game. She became traumatized and went through depression after her parents and husband died, as well as Dalmasca stolen from her. The only way she survived was through toughening up and standing up for herself, breaking out of her gentle personality into the hard one we saw in FFXII. Lightning is similar, she was also kind and calm before her parents died, but then to protect Serah, she matured rapidly and lost that.
TBH, Cloud didn't change too much throughout FFVII. If I had to pick two characters that represent Lightning, it's a 50/50 combination of Squall and Ashe.