I don't know the details of what is going on here but it sounds very sketchy. I vastly prefer an OAuth login to installing a native app, let alone one that appears to have very deep system access, and access to all my passwords at the same time.
Using the Lastpass accessibility service on Android only works about 40% of the time for me.
Over half the times I go to use it I sit there trying to trigger it a few times, then give up and have to go and copy / paste the password from the app.