That's intentionally awkward. If I tap the messages icon on Facebook.com on mobile, I get redirected to a page telling me to download the app. Similarly, messenger.com points me to the App Store.
On the contrary: You can even enable push notifications with Chrome for Android. Chat included. I haven't touched the native apps for ages and everything works just fine.
I suspect that decision to split the app comes from equal parts marketing and software engineering incompetence.