This is getting a bit over my head, so take all my opinions and guesses with increasingly larger grains of salt, and for context, my default stance is still that entering your Apple ID anywhere other than with Apple is a bad idea and I would need to be convinced otherwise.
I’m guessing Teensafe is somehow pulling the actual backup from iCloud using your credentials, and once it has that, it has pretty much everything on your phone since the last time it backed up.
If you backup your phone to iTunes on your computer, you can chose whether or not to encrypt it, but if your backups are unencrypted, there are utilities you can use for stuff like pulling your text message history out of it, and there used to be at least some GPS data saved in there as well (there was a minor scandal a couple years ago when people realized that).
So if Teensafe has a way to pull your backups from the iCloud server (and I guess get around the encryption because they have your credentials), they could get the same kind of info out of it. You wouldn’t be getting the updates in real time as your kid texts, just updates every time it backs up. I should clarify that “iCloud” is a vague term applied across several services Apple offers. Backing up your iPhone to iCloud is making a backup of all the data on your phone (though not the Apps themselves, just the data they store on your phone) to Apple’s server. It’s a separate feature from what you’re accessing when you login to iCloud.com.
I have three concerns, two mentioned already. First, is this actually malicious? On the surface, it doesn’t seem to be, and through other trusted sources and reviews, this would be the easiest of my concerns to overcome. You also have to think about the future though. If the company goes under, then what? Can someone else less trustworthy just swoop in and buy them? Do you have any protection against that?
Second, is it secure? They might have the best intentions and highest regard for your privacy, but your security is only as good as the weakest link in the chain. It’s possible anyone could get hacked/have a security breach, even Apple, but now you’re giving hackers a second target. So are you sure Teensafe is as secure as Apple is? Also, worst case scenario with Apple, if they get hacked, that’s on them, they’d be aware of it and be in the best position to make things right to the extent possible (refunding purchases, etc.). You lose some of that protection if your Apple ID is compromised elsewhere.
Third, security issues aside, if this is all being done without Apple’s cooperation (which is my assumption), it could all stop working tomorrow. If Apple changes something about how iCloud backups work, Teensafe might not be able to keep accessing them, or might not be able to get the same data, etc.
I’d never heard of services like this before, so this is all guesswork based on my “Apple Enthusiast” level of knowledge of iCloud and Apple’s systems. I’d love it if anyone with more developer background could shed some light on this.