alt Hacker NewsOn FF Android, for me, https://facebook.com/messages redirects to https://m.facebook.com/messages, and just shows a message telling me to get the Messenger app.
It seems to trigger based on the browser window being too small. Yes, it is possible to access messages by requesting the desktop site, but it's pretty inconvenient.
Using it for messaging is different. I took "the mobile app" to mean the Facebook app with the timeline, groups, etc... (com.facebook.katana on Android), which also doesn't let you access messages. They really seem to want to promote their Messenger app as a distinct product.
I'm not sure messaging can work well in a mobile browser because notifications would not be timely, but maybe you're using it for a purpose that doesn't require instant notifications.