Is Facebook Messenger Signal-based (i.e. end-to-end encrypted with a Signal/Axolotl double ratchet) as GP requested?

I somehow doubt it.

EDIT: Oh, I was mistaken, apparently there is an option now. From Wikipedia:

On October 4, 2016, Facebook deployed end-to-end encryption as an optional feature for Facebook Messenger users. It is available in an optional mode called "Secret Conversations" and uses the Signal Protocol.

Wait, are you saying messenger is secure and private? As if Facebook would want to give up a source of data for marketing?

It's opt-in end-to-end encrypted with Signal Protocol now, so Facebook does in fact give up that data. Having said that, you should prefer protocols that aren't opt-in for security, which is what Signal buys you.

It's rather a web app than a desktop application, isn't it?

It's not a desktop app, but the original requirement never stated that.

There's a desktop app for Windows. There's also a web app at https://messenger.com. You can disable your Facebook account and still use Messenger