I noticed that many beta test user agreements state that the beta tester must be 18 or older or have permission from their parent/guardian to sign up.

I know that there is a law regarding internet access and children under 13, is there a similar law for this?

I'm wondering if this just a customary procedure that the industry has just adopted, perhaps as a precautionary measure.

There is a "law regarding internet access and children under 13" in one particular country (possibly two).

The likely reason why user agreements for beta testers states that testers must be of age is that the company running the beta actually wants something.

Testing is something not as trivial as one would think, and reporting errors (in a way so it is useful) even more so. Even with educated adults that have good intent, testing is sometimes only mildly useful. I've seen entire teams who do SAP testing for a living and who have explicitly been trained on what to test in a 4 hour session fail utterly and miserably.

Now imagine a beta test with a lot of kids and teens who only play the beta because it's free and who come up with ten thousand "need more explosions" and "should be more like Runescape / Harry Potter", "need robots and ninjas" bug reports. The company running the beta wouldn't like that to happen, because sorting out the noise is unaffordable.

And then imagine that Sally Soccermom stands up because her kid played a title where the age rating is still pending, so nobody could tell what psychic damage her kid will get from a not-yet-rated game.