Each platform that supports matter has its own devices that act as “matter controllers“ for that platform and that platform only.
In order to use any matter device with a Home Automation platform, you must have one of its controller devices. These are not shared between platforms.
So in order to have matter devices show up in the Alexa app as matter devices, you must have one of the echo devices that acts as a matter controller for the Alexa platform.
In order to have matter devices show up in the Google home app as matter devices, you must have one of the Google home devices that acts as a matter controller for the Google home platform.
In order to have matter devices show up in the Apple home app, you must have a HomePod mini or an apple TV model that works as a matter controller for the Apple home platform.
And in order to have matter devices show up in the SmartThings app as matter devices, you must have one of the SmartThings devices like a SmartThings hub that works as a matter controller for the SmartThings platform.
There isn’t any way around this: it’s just the way matter works. Each matter-compliant platform has its own app and its own “matter controller” devices.
Again, see the community FAQ
FAQ: What do I need to add a Matter device to the SmartThings app? Do I need a bridge router device?