I have installed authenticator on my pixel 3 (with all of the updates) and pushes were working when I tried to access sites that I had enabled MFA. For some reason (now) none of these sites are prompting my on my phone any longer (they merely ask for the MFA code from the authenticator app) . The last time this happened, I turned off MFA for all of my sites and re-enabled it to get the pushes to work, but this is a tedious and (should be) unnecessary project.
I have been having the same issue with authenticator app on my iPhone XS. When logging into amazon and I get to the OTP the push never gets sent so I have to find authenticator app, open it, find the amazon entry tthen go back to amazon login and manually type in the OTP beffore it gets changed in 30sec! Has been a real pain in the *ss.