Algebraic Terraforming: Trees from Magma

This is really confusing to me. Let's take their definition of a free object:

"A free object over a set forgets everything about that set except some universal properties, specified by the word following free. For example, the free monoid over Integers forgets ... everything else about the Integers except: they are a set of objects, there is an associative (binary) operation on Integers, and there is a "neutral" Integer; precisely the universal properties of monoids."

This seems to be contradicted by the first statement on the following paragraph:

Now, it turns out that [a] is a free monoid of values of type a. Our first pass at constructing a free monoid might look something like this

But values of "type a" do not have necessarily an associative binary operation, let alone a neutral element. Furthermore by constructing a list of some type "a" we aren't restricting the type of a to some pre-existing ambient algebraic structure, we are creating a new algebraic structure independently of any properties of "a."

Where have I gone wrong here?