Here’s my take on this.
It’s certainly technically possible, though it doesn’t function like a simple forwarding service to your own messaging app. Most monitoring apps achieve this by requiring a one-time installation of software onto the target device. Once installed, the application intercepts copies of sent and received SMS messages and uploads them to a secure online dashboard.
You would then log into this dashboard from your own phone or computer to view the messages. While it’s often near real-time, there can be slight delays. Because this method relies on installing software, initial physical access to the phone is almost always a requirement, especially for Android devices. So it’s less about “receiving” the texts directly and more about accessing a logged record of them through a separate portal.