AWS has a very good and mature support model and is very willing to work with businesses to make their offerings fit their needs. That combined with them having home field advantage will give Azure a long fight.
Office 365 was, really, Microsoft's foot in the door to migrate people to Azure. Now, Azure AD is a foot in the door to other Azure services. And if Azure can come in and up-sell them on things like website and database hosting, with big support contracts to match, that's pretty huge.
AWS will continue to dominate among more modern businesses who have developers on staff. But I genuinely think Azure will see a lot of success servicing legacy customers who are used to the microsoft ecosystem and don't have m/any developers on staff.
All that being said; Personally: Azure is a pile of crap, with practically no redeeming qualities. I have no doubt it will do well with legacy businesses, but I really wish it wouldn't.
No doubt MS will continue to grow in this space, since it offers stability, support when things go south, as well as easy onboarding with various developer focused initiatives. While the variety of services doesn't match the AWS catalogue yet, they also offer a marketplace where 3rd party services can be purchased and integrated.