I think what I'm trying to say is that since they're all mainstream pop artists they are rivals to each other. But I do agree that people generally don't compare Namie to Ayu and vice versa since they're different in style and sound.
I remember when I started following J-pop in-detail (I grew up in Asia as a child, so I was already aware of J-pop and knew a bit of it) in my teens Namie's popularity was in her decline and people generally dismissed her as a non-factor for much of 2002 - 2005, but her popularity surged again by 2007-08 with Hide & Seek and the Vidal Sassoon promos. Then the Coca-Cola Ad with WILD was pretty much what secured Namie's place back on top, that ad was everywhere in Asia.
I do remember people did compare BoA to Namie Amuro, and as for Ayumi Hamasaki she was pretty much free-to reign since no one was exactly doing anything remotely close to her style at the time and achieving the same amount of success. There were some comparisons to Utada Hikaru, but Hikki was considered to be on her own league, her R&B-Western sound was different to Ayu's so the comparisons were limited.
Utada pretty much dominated 2000 to 2009, Ayu was a close second, while Namie was on the downlow then dominated 2008 onwards. Ami Suzuki dropped off the radar by late 2000's. BoA dominated early 2000's, fell off the radar by mid-200's. Koda Kumi has been the most consistent one to this day actually in terms of releases and sales.
Special mentions are Ai Otsuka, Yui, Maki Goto, Miwa and Kana Nishino. Ai Otsuka was huge before, she completely disappeared by the time I graduated high school! Yui was like kind of her successor in the J-pop-rock genre but she's nowhere as big compared to Ai Otsuka. Kana Nishino is by far the only new female solo-artist that is a major stand-out,, she reminds me of Ayumi Hamasaki for some reason, but more relatable (not as outrageous) and Westernized. Maki Goto was like one of my favourites but she never really took off, so much potential. Miwa is a hit and miss. Anyway, I've always wondered who will be the next-generation "J-pop Divas," and I've always consistently placed Kana Nishino as the only one so far.