There was nothing in the forum regarding 1.37 and O365 push notifications issues.
And I have no issues with the login, even though it occasionally throws a login error.
When I manually refresh the mail it works fine each time.
Scheduled sync is messed up (it works on/off) and push mail NEVER works.
I have also uninstalled Aqua Mail and installed it from scratch with not a single setting changed (so fully default) with just the O365 account, and it doesn't work, the behavior is the same.
So, in conclusion there is a bug.