No, I don't see any indication it has anything to do with the "anonymity". Very few people, even technical people care about anonymity to the extent that they try to achieve it in everyday life.

It is frequently confused with privacy, however. (https://www.privacyguides.org/en/basics/why-privacy-matters/)

iMessage is the dominant messenger because most people have iPhones combined with the fact that SMS has long been free and unlimited, so people don't see the problem of using it with the occasional Android user.

Really, it's all about the defaults. Even though everyone uses iPhones, they still use the calling feature from their cellular provider, because Apple doesn't push FaceTime as the default calling mechanism.

Signal is gaining popularity because there are people that care about using it over iMessage.

Signal is 100x better than WhatsApp, but it feels so unstable using any centralized messenger that has complete control over the software and the users. No centralized service can truly be relied on, non-profit or for-profit. But clearly that's what has to happen in order for the service to become mainstream, so it's an acceptable compromise for me. It's not like I can't say Signal does great things for privsec and metadata reduction.