Sorry about the delay. I've been testing this in my lab.
In brief, IP aliasing may not be able to work on Azure. The most similar workaround is deploying a VM with two NICs.
Actually, if you try to perform a network capture on the VM, you will find that the VM with secondary IP address is able to send out the packets. But there is no inbound traffics because Azure doesn't know the secondary address.
This screenshot is captured on a VM in the same subnet with the machine with IP aliasing enabled. 10.0.15.10 is the primary address and 10.0.15.101 is the secondary address.
According to the screenshot above, the gateway handles all network traffic (Including communication in same subnet). And that's the root cause why the secondary IP address doesn't work. Any ARP request is answered by the gateway. But the gateway isn't aware of the secondary IP, so it can't reply the ARP request for secondary IP address.
So I try to add the secondary IP configuration to the NIC, but it's not supported by default.
According to the error message, we need to register a new feature called "AllowMultipleIpConfigurationsPerNic".
When I run the command to register this feature, it is stuck in Pending state. And 2 days later, it is still in Pending state.
So, in conclusion, you may need to deploy a VM with two NICs as a workaround.
Hope these information helps.