There were a bunch of feature phones around 2014 that had an unauthorized (and poorly working) WhatsApp client. Signal of today is a bit trickier to run than WhatsApp of 2014, but it could be done. WhatsApp also ran officially on Nokia Series 40, which was a feature phone platform with apps.

There's also KaiOS which is running on phones marketted as feature phones, although it's a fork of FirefoxOS which is Android base plus an HTML+js app model (or something).

I'm not sure what you're basing this on. Java runtimes have been on feature phones for far longer than we've had smartphones. The only reason why we didn't get more capable feature phones around the time we got 3G networks is because it's the same time we got smartphones with big screens and simple touch-based UIs.

The OP seems to ask for only signal, not the possibility to install any other app. I would call that very much a feature phone.

Depends on how you use those terms. If we're talking about processing power and how sophisticated the underlying tech has to be, then yes I'd tend to agree that the ability to run modern-ish internet-connected apps is at the equivalent of a smart phone. On the other hand, if "feature phone" is about the way the user interacts - a phone that ships with a give set of features and is not intended to run arbitrary apps - then it doesn't matter how sophisticated it is under the hood, only that it bakes in certain apps/features (phone, SMS, Signal) and that's it.

The signal protocol itself should run on a microcontroller. Showing images or videos would use more compute power.