I did some testing with the new Android app. Before it would save my M+B ID, but not password. Now, when I go into settings and add them, it just logs me in every time. So I can start the app, hit the blue "Miles + Bonus" button at the bottom and I'm in.

To test it, I manually logged out using the "Logout" button. Then shut down the app (swiped it off in the task switcher. Haven't tried a force close). When I restarted it (splash screen and everything) I could hit "Miles + Bonus" and I'm in again. No logging in.
EDIT: Apparently after a reboot now it doesn't automatically log me in... but it does save login/pass. I swear I was not making that up before. Completely different behaviour. FWIW Nexus 5 running 6.0.
I wonder if it just wasn't fully closing before (e.g. swiping it away didn't really stop it)? Oh well. Still saves password data. One step up from before.

Only complaint is that I had to do this "backwards" from what I'd expect. E.g. I had to add all my data in the settings so I could get into M+B, when I would have expected rather that it populates my data from my M+B account.... but hey, it works now. Big fan of the new app.