iOS is a sandboxed operating system, and thus third party developers, like MS or Google, do not have the ability to write apps that access certain core services on an iPhone. The iOS Message App handles SMS/MMS and Apple iMessages, and since it is Apple’s code, they can build their continuity feature into their operating systems so an iPhone, iPad and Mac can share and sync messaging. But no third party app, like YourPhone, has any access to the iPhone iOS Message app to sync data with none Apple operating systems or devices.
The whole aspect of a sandboxed operating system and closed, Apple controlled architecture is at the very heart of Apple’s promise to build secure mobile devices. Obviously that doesn’t work for everyone, but is one of the reasons many of us choose not to use other operating systems for our personal devices.