On iOS there really isn’t a “set-and-forget” app that lets you pull full Messenger chats or call logs without either jailbreaking the phone or getting its iCloud credentials and running a desktop backup-parser—both of which void warranties and can brick the device. Your most straightforward (and Apple-approved) option is Family Sharing + Screen Time: set up your child’s device in your family group, enable Screen Time limits on Messenger, and mirror their incoming notifications to your own iPhone, iPad or Mac so you see message previews in real time.
If you did want deeper access you’d have to jailbreak and install something like FlexiSPY or mSpy, but that’s risky and Apple will patch it ASAP. On Android you’ll find more “spy” apps that claim full chat/call access, but they almost always require root, can introduce malware, and are anything but reliable. In practice, open communication plus Screen Time controls on iPhone is the safest, most stable way to keep tabs on Messenger use.