Learn English – An expression for comparing things accurately in English

The term honey pot is often used for this.

A sweet, tempting attraction to trap an attacker.

First of all, I wouldn't call it a commonly-used idiom, although it would be hard to show that, because the phrase in the soup can be used in a lot of different ways (cooking being the most obvious). I did a Google search for "I am in the soup," and didn't find all that many hits. Moreover, even ones that looked promising didn't mean necessarily mean "in trouble"; for example:

the pilot again stated that he was "in the soup"

I thought that might mean that the pilot was in trouble, but, looking more closely at that webpage:

When interviewed later, the controller stated that he did not realize that "in the soup" meant "in the clouds."

Second, even though a dictionary can verify that in the soup can mean in trouble, I don't think that's what the author is conveying here. I believe the soup here is simply a metaphorical broth, and:

The Internet is the soup that our students have been swimming in

is about the same as:

The Internet is the world that our students have been living in

as evidenced by the author's prior statement:

Children are not going to miss out on the glories of Facebook and YouTube because teachers failed to incorporate social media in their classrooms. The Internet is the soup that our students have been swimming in all their lives.

In other words, the Internet is all around them.