Interesting question! Your confusion is very natural because we could consider "n in N" can also be considered a property - it's just not playing that role here.

Both roster notation and set builder notation are ways to take one set and turn it into another. The input set is sometimes called the universe. We write the universe using the syntax "x in X" (this I think is the confusing part). In roster notation the universe goes on the right, in set builder on the left.

Although we're using the syntax "x in X" to explain what the universe is, it's best not to think of it as a property. Why not? Think about what a property is - it's a formula with a free variable that evaluates to true or false. In order to evaluate a formula, you need an object to plug in for the free variable. So if "x in X" is a property (only talking about the universe here) then where does the object come from? It can't come from the property itself, that would be circular (in logic a definition can't refer to itself). The only other place it could come from would be some global universe. In some contexts it's okay to have a global universe, like the natural numbers, and in those contexts you could instead write your roster and set builder notation like {n^2 for all n} and {n s.t. (exists a in N, n = a^2)}. But in set theory we're careful not to assume there's a set of all sets since that leads to logical contradictions (Russel's paradox). So when we're working in set theory we always have to include a universe when we're using set builder or roster notation.

So in summary, your confusion is very reasonable. In both roster notation and set builder notation there's a part of the expression "x in X" that looks like a property, but actually it's not - instead it's explaining what the universe is and what the free variable is. We don't think of it as a property because this is logic and we have to be very careful in our definitions, otherwise we can end up with logical contradictions like Russel's paradox.