The Avril comparisons are a really hard sell for me honestly. Avril was maybe pushing a countercultural image, but that's where the comparisons really end. Growing up when Avril's debut hit, her whole shtick felt like it was consciously designed to be a commercially digestible version of teen angst positioned to maximize sales at Hot Topic. In short, it never felt authentic. A former country singer whose songwriters and marketing team designed to be countercultural but in that way that always felt hollow to me.
Eilish however, no matter how you feel about her music (by personal opinion varies from song to song), has never felt in authentic. The entire album was written by both her and her brother, not a song writing team trying to develop a pop star. Her image is driven both out of self expression and a desire not to be objectified in the way most female pop stars are.
If I had to make a comparison, I would liken Eilish more to Fiona Apple. It feels a lot more adept of a comparison, especially in the song writing.
Oh thanks for pointing this out since I didn't know that Avril didn't write her all songs at all. I've just checked some sources in the internet and it seems that we can't actually make a comparison in terms of songwriting, you are right.