What is the Microsoft Azure equivalent to Amazon SES service?

There is nothing built into Azure. You must consider third-party services.

A popular one is SendGrid (Tutorial | Volume Pricing) but there are definitely other options. FYI, I've not used SendGrid myself for anything with heavy volume so I can't really recommend them based on my personal experiences, but they seem to be one of the most popular go-to options for this area.

For low-volume SMTP, I've used Office 365 for quite some time myself.

Another options is, you can use Amazon SES! Scott Hanselman has a neat blog post about this option.

Lastly, I stumbled upon this really neat post called "Transactional Email Service & Bulk Email Software". I absolutely cannot vouch for the reliability of the information there but it seems like it could be a great tool and it was posted/last updated within the past two weeks, so it's really worth checking out!

Windows Azure itself does not natively provide anything like that, but SendGrid is in the Windows Azure Store and can be very easily integrated into your Azure services.

I came here hoping there was a new solution in 2019, but still bad luck.

After using SendGrid for a while I switched to Amazon SES - SendGrid is badly integrated and very expensive compared to SES on larger quantities. I cut my cost by 10 switching to SES.

Pretty crazy that Azure force people to Amazon for something so basic as an SMTP service.